JOEL - I did a search of all the songs in the Hymn book that contained the word "pride". Here are the results:

"And quell the sinner's pride" - Behold, the Mountain of the Lord - Hymn #54
"Pride ruled my will, remember not past years" - Lead, Kindly Light - Hymn #97
"More pride in His glory" - More Holiness Give Me - Hymn #131
"Land of the Pilgrim's pride" - My Country, 'Tis of Thee - Hymn #339
"How can it please the human pride" - Nay, Speak No Ill - Hymn #233
"The flowers on every side, Blooming in stately pride" - Our Mountain Home So Dear - Hymn #33
"Then cease from all light speeches, light-mindedness and pride." - Ye Who Are Called to Labor (Men) - Hymn #321

Most of these songs are warning us against pride and how sinful it is. One is talking about flowers and another is a National hymn. Hymn #130 is called "Be thou Humble" and does not contain the word pride.
Pride is sinful because it usually directs attention on ourselves or puts others down to make us look better than them (Jacob 2:13). The phrase in hymn #131 ("more pride in His glory") is putting all the attention on the glory of God, not on ourselves. It encourages us to do all we can to give all the glory and honor to God. Besides this I think "pride" is the only word that could fit with the notes of the song best. Any other similar words would have had more than one syllable.
We need to consider the context of how and why a word is being used in a song before getting too worried about it (especially if it has to fit the melody of a song or rhyme with something).
For example in the Book of Mormon we find a scripture that says, "Wo be unto him that crieth: All is well! (2 Nephi 28:25) Yet in the song "Come Come Ye Saints" (#30) we sing the phrase "All is well, all is well."
In the Nephi scripture it is speaking against the people who think that they can follow satan and still feel secure in their salvation. Verse 21 says:

"And others will he pacify, and lull them away into carnal security, that they will say: All is well in Zion; yea, Zion prospereth, all is well—and thus the devil cheateth their souls, and leadeth them away carefully down to hell." (2 Nephi 28:21)

While in the song, the righteous pioneer saints sang these words to give them hope and comfort in their times of suffering.

In regards to pride itself; I don't think there is any "good" pride; just different degrees of bad pride. There is nothing wrong with having a healthy sense of self-respect or self-worth, but in the true scriptural deffinition of the word, there is no such thing as righteous pride—it is always associated with sin (Ps. 59: 12 , Prov. 21: 4 , Mosiah 29: 9 , D&C 98: 20).
Pride places enmity between us and other people or between us and God.

C. S. Lewis said,
“Pride gets no pleasure out of having something, only out of having more of it than the next man. … It is the comparison that makes you proud: the pleasure of being above the rest. Once the element of competition has gone, pride has gone.” (Mere Christianity, New York: Macmillan, 1952, pp. 109–10.)

Pride is the power by which satan gets control over us (3 Ne. 6: 15) . Pride is the antonym of humility. Both cannot exist in the same place at the same time.
One might ask, "Is it wrong to tell my child how proud I am for what he has done?" Something like this, said in private, might indeed instill a feeling of love and approval in a child from his parent, and probably isn't all that harmful, unless of course the parent is having feelings of superiority over other parents, who's children are not so talented.
It is difficult to refrain from using the word "pride" because we have been conditioned in this world to regard it as a good thing. We are proud of our team; we are proud to be an American; The Few, The Proud, The Marines! etc.
But what can start out as a seemingly harmless expression of love and approval can, if we are not careful, lead to feelings of superiority and arrogance over others.
Perhaps in cases where one might mean to convey feelings of love and happiness for another person, or feelings of self-respect or self-worth for ourselves, it might be more appropriate to use the word "pleased" rather than "proud".