Definitions:

STRONG:  Middle voice from LOgos (3056); to take an inventory, that is, estimate (literally or figuratively).

THAYER:  

1. To reckon, count, compute, calculate, count over.
   1. To take into account, to make an account of.
      1. Metaphorically to pass to one’s account, to impute.
      2. A thing is reckoned as or to be something, i.e. as availing for or equivalent to something, as having the like force and weight.
   2. To number among, reckon with.
   3. to reckon or account
2. To reckon inward, count up or weigh the reasons, to deliberate.
3. By reckoning up all the reasons, to gather or infer.
   1. To consider, take into account, weigh, meditate on.
   2. To suppose, deem, judge.
   3. To determine, purpose, decide.

VINE:  To reckon.

1. To think. (Rom. 2:3)
2. In 1 Cor. 13:5, "taketh (not) account of" (RV), i.e., love does not reckon up or calculatingly consider the evil done to it (something more than refraining from imputing motives).
3. In Phil. 4:8, "think on (these things)", it signifies "make those things the subjects of your thoughtful consideration", or "carefully reflect on them". To think upon a matter by way of taking account of its character.
4. To deliberate, and so to suppose, account.
5. In 2 Cor. 3:5 the Apostle uses it in repudiation of the idea that he and fellow-servants of God are so self-sufficient as to "account anything" as from themselves, i.e., as to attribute anything to themselves.
6. To put down to a person's account. In 2 Tim. 4:16 it is used of laying to a person's "account" as a charge against him.

ZODHIATES:  To reckon, impute, number.
Actually, the verb means to put together with one's mind, to count, to occupy oneself with reckonings or calculations.

In the New Testament,
1. The present used in a passive sense.
2. Also means to reckon, count.
3. To reason.
4. To think.
5. To count something to somebody means to reckon something to a person, to put to his account, either in his favor or for what he must be answerable.
6. To reckon, to value or esteem.
7. To account, to conclude or infer, to believe.

References:

1. Rom 4:4,5,24; 9:8
2. (1 Cor 13:5)
3. Mk. 11:31
4. Rom. 2:3; 8:18; 1 Cor. 13:11; 2 Cor. 3:5; 10:7,11; Phil. 4:8; 1 Pet 5:12
5. Rom. 4:3,4,6,8,9,11,22; 2 Cor. 5:19; Gal. 3:6; 2 Tim. 4:16; James 2:23.
6. Rom. 8:36; 1 Cor. 4:1; 2 Cor. 10:2; 12:6; Rom. 14:14; 2 Cor. 11:5; Phil. 3:13; Heb. 11:19; Rom. 6:11.
7. Rom 3:28.

Definitions:

1. Actively causing evil, i.e. hurtful, harmful;
2. In the neuter, evil, i.e. cause or source of evil;
3. Evil done to anyone, harm, injury;
4. In words, evil speaking;
5. Pl. evils, i.e., troubles, affliction.

References:

1. Rom. 14:20; Tit. 1:12; Rev. 16:2; Sept.: Prov. 16:4; Amos 6:3;
2. James 3:8;
3. Acts 16:28; 28:5; Rom 12:17,21; 13:10; 1 Thess 5:15; 1 Pet. 3:9,11;
4. 1 Pet. 3:10; Sept.: Mic. 7:3;
5. Lk. 16:25; Acts 9:13; 2 Tim. 4:14; Sept.: Gen. 44:34; 48:16; Is. 46:7; Jer. 14:8.

From Ancient Christian Commentary on Scripture: Romans

Origen (c.185-c.254)--If some people think it is wrong to do evil but not wrong to repay it, it may be just but it is still a similar sin, or in my opinion, even a worse one. For the one who does evil to begin with may perhaps not realize that what he has done is wrong. But the one who repays evil and who is moved by thoughts of revenge has already admitted that he knew it was wrong to do it. (cf. Rom 12:17)

Ambrosiaster (fl. 366-384)--Evil works against itself, and when it is overcome it thinks that it has won! Our enemy acts in such a way as to divert us from our purpose, looking for an opportunity to make us sin. Therefore if we are provoked by him and do not reply in kind, we overcome him with good. Therefore we do not resist, ... (cf. Rom. 12:21)

Chrysostom (c.344/354-407)--Paul means this: As far as possible, play your part and give nobody, either Jew or Gentile, any cause for fighting. But if you see the faith suffering anywhere, do not put harmony above truth. Make a noble stand, even to the point of death. (cf. Rom. 12:17)

A writer tells how "in Polynesia, where the natives spend much of their time in fighting and feasting, it is customary for each man to keep some reminders of his hatred. Articles are suspended from the roofs of their huts to keep alive the memory of their wrongs--real or imaginary." So many people nurse their wrath to keep it warm; they brood over their wrongs until it is impossible to forget them. Christian love has learned the great lesson of forgetting.

St. John Chrysostom, Homily 33:  And he said not, "worketh no evil," but, "not even thinketh;" i.e., so far from contriving any evil, she doth not even suspect it of the beloved. How then could she work any, ...who doth not even endure to admit an evil surmise[?]

Adam Clarke:  Thinketh no evil] oulogizetai to kakon? "Believes no evil where no evil seems." Never supposes that a good action may have a bad motive; gives every man credit for his profession of religion, uprightness, godly zeal, &c., while nothing is seen in his conduct or in his spirit inconsistent with this profession. His heart is so governed and influenced by the love of God, that he cannot think of evil but where it appears. The original implies that he does not invent or devise any evil; or, does not reason on any particular act or word so as to infer evil from it; for this would destroy his love to his brother; it would be ruinous to charity and benevolence.

John Gill's Exposition of the Bible:  thinketh no evil; not but that evil thoughts are in such a man's heart, for none are without them; though they are hateful, abominable, and grieving to such as are partakers of the grace of God, who long to be delivered from them: but the meaning is, either that one possessed of this grace of love does not think of the evil that is done him by another; he forgives, as God has forgiven him, so as to forget the injury done him, and remembers it no more; and so the Arabic version reads it, "and remembers not evil"; having once forgiven it, he thinks of it no more; or he does not meditate revenge, or devise mischief, and contrive evil against man that has done evil to him, as Esau did against his brother Jacob; so the Ethiopic version, by way of explanation, adds, "neither thinks evil, nor consults evil"; or as the word here used will bear to be rendered, "does not impute evil"; reckon or place it to the account of him that has committed it against him, but freely and fully forgives, as God, when he forgives sin, is said not to impute it; or such an one is not suspicious of evil in others, he does not indulge evil surmises, and groundless jealousies; which to do is very contrary to this grace of love.

John W. Gregson:  "thinketh (logizetai) or considers no evil;"

"Love often thinks of evil. To be otherwise is to be naive. But love does not spend all of its time in thinking about evil". ... Love thinks no evil; it does not keep a ledger of wrongs to repay later.

Matthew Henry:  Charity thinks no evil. It cherishes no malice, nor gives way to revenge: so some understand it. It is not soon, nor long, angry; it is never mischievous, nor inclined to revenge; it does not suspect evil of others, ou logizetai to kakon--it does not reason out evil, charge guilt upon them by inference and innuendo, when nothing of this sort appears open. True love is not apt to be jealous and suspicious; it will hide faults that appear, and draw a veil over them, instead of hunting and raking out those that lie covered and concealed: it will never indulge suspicion without proofs, but will rather incline to darken and disbelieve evidence against the person it affects. It will hardly give into an ill opinion of another, and it will do it with regret and reluctance when the evidence cannot be resisted; hence it will never be forward to suspect ill, and reason itself into a bad opinion upon mere appearances, nor give way to suspicion without any. It will not make the worst construction of things, but put the best face that it can on circumstances that have no good appearance.

Jamieson, Fausset, & Brown:  thinketh no evil -- imputeth not evil [ALFORD]; literally, "the evil" which actually is there (Pr 10:12; 1Pe 4:8). Love makes allowances for the falls of others, and is ready to put on them a charitable construction. Love, so far from devising evil against another, excuses "the evil" which another inflicts on her [ESTIUS]; doth not meditate upon evil inflicted by another [BENGEL]; and in doubtful cases, takes the more charitable view [GROTIUS].

BW Johnson:  Thinketh no evil. The idea of the Revision is that love does not keep a record of evil rendered so as to return it.

Mark Heber Miller:  (Love) does not keep account of the injury.

The Greek is OU LOGIZETAI TO KAKON, literally, "does not keep record of wrongs." Or, "keeps no log on bad things." One can see the English word "log" of "logistics" in the Greek root. The phrase is variously rendered: RSV: (not) resentful; KJV: thinketh no evil; MON: bears no malice; TCNT: never reckons up her wrongs; NEB: love keeps no score of wrongs; PME: does not keep account of evil; NAS: does not take into account a wrong suffered; WMS; never harbors evil thoughts; BECK: it does not plan to hurt anyone; NJB: does not store up grievances.

The Proverbs taught to "pass over transgression." (Proverbs 19:11) The first occurrence of "forgive" in the Bible is that of God. (Exodus 34:7) The Psalmist describes God as "ready to forgive" and the Prophets describe Him as "forgiving in a large way." (Isaiah 55:7)

Perhaps this is one of the hardest challenges in showing love: not only forgiving but also forgetting personal injuries. The Nazarene taught us to pray, "Forgive the debts of others." (Matthew 6:9) Or, let go, relieve those indebted to us. Then, Jesus makes the first commentary on the need for forgiveness when he follows with: "For if you forgive others their trespasses your heavenly Father will forgive your trespasses." (Matthew 6:14) Our Lord puts it simply: if we refuse to forgive others we cannot expect forgiveness from God. (Mark 11:25)

The Nazarene answered the disciples’ question on how many times we ought to forgive. Peter suggested as much as seven times per day. The Lord said, "Up to 77 times" then gave an illustration of such forgiveness in the context of financial indebtedness. (Matthew 18:21-35) Luke 17:3, 4 adds a proviso to this forgiveness: the offending person must come and say, "I am sorry." Paul elsewhere counsels "freely forgiving." (Ephesians 4:32; Colossians 3:13) Even if a sinner has brought the congregation into disrepute and has repented, all are to forgive him or her. (2 Corinthians 2:7, 10)

Some persons go through life with a little book in their head. This log contains all the injuries they have suffered at the hands of others. In a heated argument they will bring out this book and recite chapter and verse of all the wrongs done against them. These persons are not loving and therefore really hateful. The mature Christian will be characterized by a forgiving disposition who truly forgets offenses or sins committed against the person. What is helpful in doing this is not taking oneself so seriously and emptying self of egotism.

Robertson's Word Studies:  {Taketh not account of evil} (ou logizetai to kakon). Old verb from logos, to count up, to take account of as in a ledger or note-book, "the evil" (to kakon) done to love with a view to settling the account.

BT Internet:  logizomai, reckon, count, credit, suppose

Either, 'does not keep a record of wrongs', or, 'does not think evil'--i.e. always imputes best motives. Bruce thinks the reference is to determination to pay someone back in their own coin.

kako", h, on, evil, bad, wrong, harm

The Theologian: The Internet Journal for Integrated Theology:  Ou logizetai to kakon--Touchiness leads to the making of lists and an unforgiving attitude. This would hit home forcefully to a Church like the one in Corinth, which was riddled with factions, each no doubt taking note of how the others reacted. "The man who loves does not compile a dossier about his neighbour."

Bill Turner:  Love thinks no evil, is not mindful of wrongs.

"Logizetai," is the present middle passive indicative of "logizomai," to count, to set down as a matter of account, as in a ledger. Love does not keep an account of wrongs done to it, or scores to be paid back. Love does not have outbursts of anger, or keep that anger burning, by cherishing real or imaginary records of the failures and sins of fellow Christians. Christians with "agape love will love their enemies, even if they do not like them or their ways; and prays with genuine concern for those that despitefully use it. Mt.5v43-48. Lk.6v26-28. 23v33,34. Acts.8v51-60. Love does not compile or settle accounts with people. It finds no sense of superiority or satisfaction in the faults of others. Those who harbour a bitter unforgiving spirit can expect severe discipline from God. Mt.5v21-26. 18v21-35. Mk.11v25,26.

Injuries and emotional wounds caused by hurtful words and actions should be dealt with immediately and not allowed to fester, and where there is repentance there must be whole-hearted forgiveness. Mt.5v23,24. Lk.17v3,4. If the Church fails to follow Christ's procedure of discipline laid down in Mt.18v15-22., then the Lord Jesus personally disciplines His loved ones. 1Cor.11v27-32. It is better to put matters right now than to have to do it at Christ's judgement seat. Rom.14v10-13. The injured party should be humble and perform positive acts of kindness to the person who has injured them, so that repentance and reconciliation is made much easier. Gal.6v1. Rom.12v14-21. 1Cor.1v10. Phil.2v3. Eph.3v8. 1Tim.1v15.

Vincent's Word Studies:  Thinketh no evil (ou logizetai to kakon). Lit., reckoneth not the evil. Rev., taketh not account of. The evil; namely, that which is done to love. "Love, instead of entering evil as a debt in its account-book, voluntarily passes the sponge over what it endures" (Godet).