Blue Angel is a short cab ride away and the shore dive there is pretty good. Tanks and weights there are about US$6, I believe. The restaurant there is excellent and stays open late enough for you to eat there after a night shore dive. The dive shop closes at 5, though, unless they are running a night boat dive.
One thing: for shore diving without a guide, especially at night, I highly recommend taking a compass because it's easy to get turned around out there, and if the current changes direction on you, which it does sometimes, you could end up heading out to sea instead of into shore if you are using the current as a reference. Those cheap fluid filled hiking compasses on the plastic plates work just fine.
If your eyes are better than mine maybe.
A few years ago I was there with my home bud who never was one to think about details on a dive. I'd promised his mom several times that I'd always make sure he came back alive from our dive trips. (He's pushing 60 now I think, I'm older, his mom is older yet - but still.) It was last time staying at the Blue Angel and we did a nighttime shore dive around the stingray pen, then southerly along the shore, heading into the current. About the time we got close to the steps where some locals have laid sand bags over the ironshore, and I thought we should turn around to head back - the current changed. It's not the main Cozumel current that is almost always south to north more or less, part of the great
North Atlantic Gyre, but close shore eddies that come and go.
Anyway, we stood up near the stairs, looked back towards the hotel, and I told him to shoot his compass for the sting ray pen and we'd swim back to it, then around to crawl out. I know that he is also lousy about following me, so I was planning on following him to make sure we both got there and not heading out in the channel. He looked at his compass for a minute - in the dark, then asked
"How does this thing work again?"
I said,
"See the stairs? Go thataway!" We walked back to the hotel.
It's a little over a mile to Blue Angel where you can rent tanks & weights. I think you'll find better shore diving just south of there.
As Gordy said, they have a nice cafe there. You can leave your dry bag with the shop; just make sure you get back before they close. Money hidden in the bag will probably be safe enough. Cozumel's low crime rate does seem to be growing, but it's still low - especially for tourists. Most prefer to protect tourists so they can overcharge the masses as much as possible rather than steal from them and give the place a bad name. Cabbies are the most common about overcharging. Most seem pretty nice, but enough aren't - so agree on the price each way before you ever board one. It's also nice to be dry and in dry clothes, or you may not get a cab.