There are so many great hymns it is hard to narrow it down. I have lead the music in various churches for about 24 years and the first 20 or so it was always hymn style singing. So, I have used a lot of beautiful hymns over the years.
Several that have already been mentioned are among my favorites--Be Thou My Vision, How Great Thou Art, etc.
Here are some of my very favorites that have not been mentioned yet:
"Others" Click the the title to see the lyrics and hear a midi version... they take it too fast... by the way
"Living for Jesus" I found an interesting version on YouTube for this one.
"Now I Belong to Jesus" Another video with 3 men singing it A Capella... Not bad harmony.
"He Lifted Me" Another video... she does a decent job with this but a little fast I think. No doubt she's nervous.
There are many others but these are some of my favorites.