I've been using Marina Divers (
http://www.reefdivinggroup.co.il/eng/) for years. Always had a good service; they provide guided dives in groups, boat dives to couple of sites that are restricted for shore access and also help single divers to find buddies if they wish to dive by themselves and not in a group. I like their location - right of their shore is a very nice wreck with bottom touching 100ft (30m) with some very nice coral area if you pass the wreck and explore just further out.
Another good option, just a bit further south is Aqua Sport (
http://www.aqua-sport.com/), one of the oldest centers in the area which I used in early 90s. They also have a little wreck off their shore but if you walk a bit south, you easily dive into reef preserve at shallow depth. They also should have boat options for a day to remote destinations and guided group dives along the shore ( transportation provided).
There are many dive centers in Eilat but as far as my experience, each one will have more than decent rental gear and they all will help you with recommendations for diving spots if you have your own transportation.
As for hotels, I will let someone local to address that.
One last thing, Israeli diving federation passed local directive that requires each diver to carry dive insurance ( could be purchased locally, DAN is accepted) and to have logged and signed dive within at least six months prior to you diving day; otherwise a refresh check-out dive is required.
Enjoy your visit and dive safe
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