Well, sticking to Beethoven to start, try 'Fur Elise'. It's short and not quite in the same temperment, but you may like it. In fact, it's rather hard to match temperment (At least for me) in classical, so you may find you don't care for any of my recommendations, but good luck.

Then try 'Claire De Lune' by Claude Debussy, which you'l recognize as infinitely abused in advertisements.

'Pavane' by Gabriel Faure is another possibility, although the pacing is faster it's still somewhat melancholic.

'Canon' by Johann Pachelbel is worth a listen, although the mood it evokes is more difficult to describe.

You should also look into opera, as you'll find plenty of melancholy there.

If you're interested in more powerful, grim sort of music, one of my favorite pieces is 'Totentanz' by Franz Liszt. Also in that vein 'A Night on Bald Mountain' by Modeste Mussorgsky.

Don't limit yourself to one style of classical either, try it all out. Oh, and I'm more of an orchestral fan than a pure piano fan, so you might want to ask someone who plays piano for their recommendations. Solo piano pieces are more likely, I would guess, to have that sort of tone you're seeking.