I'll drop my two cents in for Wonder Boy 3: The Dragons Trap. It's not system exclusive and it uses passwords but man, it's a pretty easy game to get into. I liken most to Metroid style although it's far more playable in this day and age. Lots of areas to explore, good controls, and great visuals all lend themselves to the experience being enjoyable. I began playing it this past year and I was pleasantly surprised.

Power Strike (Aleste) is also quite good if you can track a copy down. Thankfully, the European release is much cheaper (and easier to find) if you source a cartridge only copy. Power Strike 2 however is very expensive/elusive although it's an excellent game, so I'd try before you buy on this if you're unsure.

I also quite enjoy Shinobi and Choplifter, which are both cheap and easy to find.