The Reverend Mommy commented down below on one of "The Faith We (shouldn't) Sing" entries that it would be good to compile a list of theologically sound songs; I tend to agree with her implication that that that might be a shorter and easier list to come up with!
A song that's growing on me lately (no, not like moss) is "Blessed Be Your Name." You can link to the lyrics here.
First of all, while the song is simple musically, it incorporates a wide enough melodic range and interesting rhythm so that it does not become boring. Its form includes three distinct "sections" that contrast with each other and add to the musical interest.
Secondly, this song quotes or paraphrases Scripture extensively, but with cohesiveness. Matt Redman kept close to his theme of praising God at all times, in all circumstances.
Third (and this is a huge plus with me), the song is solidly second-person, directed at God. It's a song TO God, not ABOUT God. Paradoxically, I think it tells far more truth about who God is, than any third-person P&W song I know of.
The one thing I wish could be changed about this song is for it to be a corporate text, rather than individual. When we gather for worship, at least some of our song should be from US, the gathered Body, rather than all be from ME. Frankly, the dearth of corporate texts means there are weeks when our "contemporary" service has no corporate songs. This is one of the problems with the individualism that plagues evangelical-dom, out of which P&W-dom comes. But when the text is otherwise as good as this one, this shortcoming is not a deal-breaker.