I disagree with the idea that most NES games haven't aged well. I think part of the problem is nostalgia. Most games don't hold up to your memory of them. I, however, didn't own an NES as a kid, only played games once in awhile at a friend's house. So here's my list of my favorite NES games, unclouded (mostly) by nostalgia:
1) Dragon Warrior - Yes, this RPG is fairly rudimentary. What do you expect considering when it came out? But it does a lot of things well. The best aspect is that the world feels completely open, like you can go anywhere you want if you can only survive the beating that the enemies will pile on you. That's something that most RPGs lack.
2) Crystalis - This is the best action-RPG title on the NES, and one of the best of all time.
3) Legend of Zelda - Do I need to say anything about this game?
4) Kirby's Adventure - This is my favorite platformer on a system that is known for its excellent platforming games.
5) Super Mario Bros 3 - Ok, this is really not "my" pick, I suck at platformer games so most of them don't hold my interest as far as playing goes, but I love watching other people play this game, and it really holds up well against ANY game you put it up against. It's arguably the best looking game on the NES.
6) Mike Tyson's Punch-Out!!! - Holy cow is this game FUN! I hate sports games, I'm not big on action games, I don't like puzzle games... this game is all that but it does it in such a way that I can't help but keep playing it over and over again. A true classic.
7) Abadox - There are several good shmups on the NES, but this is the one I love. It is serious gory fun. Again, I'm not a fan of shmups, but I love this game.

Arkista's Ring - Excellent puzzle/adventure game. Really fun.
9) Destiny of an Emperor - This game was made back before developers settled on a way that RPGs should play, back when experimentation was still the name of the game. DoaE mixes RPG and strategy elements in ways that have never been done since. Instead of hit points, you have army men. Your party consists of generals who, when attacked, lose men, not health. There are other strategy bits in the game, but I don't want to give too much away. Definitely give it a try.
10) Legend of Ghost Lion - You can tell I'm an RPG addict, can't you? This is another game that turns RPG conventions on its head. Its super cute, full of self-help-ish messages, and still features very rudimentary gameplay. But for all that it's a great game. Instead of having a party of adventurers, you have items that when used in battle summon creatures to help bttle enemies. You don't have hit points, magic points or experience; you have hope, dreams and... something out equally cute, I don't remember. You also don't level up from grinding away at random battles; instead you get pieces of hope that cause you to level up. This makes the game quite a bit more difficult, in a good way.
Lots of other people will mention Castlevania, Metroid, Final Fantasy and things like that. I either don't like the games or haven't played them, so I can't comment on them. Same with the rest of the Dragon Warrior series. I'm playing DW2 right now and working my way thru them, so I can't say if the others are better and deserve to be on this list.