J. H. Jowett, Brooks by the Traveller’s Way:  “Love seeketh not her own.” So far from rushing into any unseemliness in seeking to display itself, so far from trampling upon the rights of others, love does not even claim her own. “Love seeketh not her own.” She claims no rights except where moral principle is involved, and on this she takes a stand, and the gates of hell cannot prevail against her. There is a quaint, grey monument in the sweet old town of Appleby, which was built in the days of the Puritans, and on which these words are inscribed: “Maintain your loyalty; preserve your rights.” Maintain your rights! Aye, but they were the crown rights of manhood, freedom to oppose iniquity, freedom to worship God, and the very love in the hearts of those strong old Puritans made them claim the rights, and support their claim by death. There are rights which true love will never relinquish. She will always seek her own. On the other hand, there are rights which love is ever prepared to yield to others. If love had a right to the uppermost seat at a feast, and somebody else has got it, love would seek not her own, but would gracefully insist on the rights of the other. If love had a sitting in the Church of Christ, and came and found that someone else was seated there, love would not behave itself unseemly; love would seek not her own, but would cheerfully seek a seat elsewhere. Is not this the way of love? Would not this be the way of Christ? How many opportunities there are, in the whole round of life, where love might graciously abdicate its own rights for the comfort and interest of others. Let us keep our eyes open, that when the Master gives us such opportunity, we may use it according to His desire. And, some day, when the evening of our life is come, He will come to us, and because we have sought not our own, but have cheerfully yielded to others, He will whisper to us, “Friend, go up higher,” and the word will make us leap for joy as we enter the eternal world. “Love seeketh not her own.”

Steve Lewis: 
It does not seek its own (zeteo) = to seek one's own interests as the primary concern. Paul used this same term when he made a similar statement in 1 Cor 10:24 -- "Let no one seek his own, but the good of his neighbor." There Paul explicitly stated what a man was to seek, rather than seeking after his own interests.

Mark Heber Miller:  (Love) does not look for its own interests.

The Greek is literally "not seeking things of itself." (OU ZETEI TA HEAUTES) It is variously rendered: KJV: seeketh not her own; MOF: never selfish; RSV: does not insist on its own way; TCNT: never self-seeking; NOR: not pursue selfish aims. Perhaps no phrase describes the general understanding of agape-love. The idea is expressed elsewhere by Paul. Indeed, a similar phrasing in Greek has already occurred in 1 Corinthians 10:24, "Let none seek selfish interests, but rather the interest of others." Philippians 2:4 is very similar: "Do not be looking after selfish interests, but rather those interests of others."

Here is the root of agape-love: interest, not in self, but in that of others. Truly, this is neighbor-love characterized by the Golden Rule: "Do to others just as you would have it done to yourself." (Luke 6:31) This means putting others before self, just as the example of our Lord, "Though he had a divine existence he did not insist on retaining his own rights, but rather he emptied himself and took on a slave’s existence in the likeness of humankind." (Philippians 2:6, 7) This is Paul’s example of not looking after just one’s own interests as he mentions in Philippians 2:4.