Cozumel doesn't see as many young travellers (at least if you limit yourself to actual DOB) as places such as Playa del Carmen or Isla Mujeres.
Still lots of fun to be had if you don't expect crazy parties and don't mind hanging out with an older (again, only referring to DOB) crowd. I've stayed twice at the Hotel Pepita (on 15 Avenida, two blocks from the main plaza), no ocean views but a nice patio (free coffee in the morning, free water and free wifi connection there) where you can meet other clients. It attracts a mixed crowd (backpackers, older single divers, local families) and is a pleasant place to stay at a good price (about 35 USD a night).
I've also heard that the hotel Barracuda can be a fun place if you don't mind a bit of noise/music in the afternoons as the cruise ship crews hang out there. But I've never stayed there, so can't comment more.
You can meet some cool and fun people in the bars in the evening. Good spots (live music) include Money Bar (South of town, you will need to take a cab) on the week-ends, La Hatch (in front of Villa Blanca hotel) and La Palapita next door, Love Café Cozumel (on the beach just north of town), French Quarter (in town). Ambar and Piratas (5 Avenida half a block south of the plaza) also fill up with local youths when they have live bands playing. There are also places like Hard Rock Café, Carlos and Charlie and Señor Frogs.
If you are flying out of Cancun and really want to insure some heavy partying (AFTER you're done with all your diving of course) with people your own age, then I suggest spending the last night (or 2) of your trip in Playa del Carmen (reknowned for its parties) or Isla Mujeres (much smaller and lower key, but filled with backpackers, so easier and faster to meet people).
