Since the OP wants to live, work, and dive for a month, I'm thinking the area north of 7MB on the West side of GC might be a good option because cheap and easy shore diving will be readily available. A month of boat dives in the Caymans could get expensive - although I don't know the size of the OP's budget - but if they are used to traveling to Mexico then they may be looking for something more economical.
@beth_castroll Where are you traveling from, what time of year are you planning to travel, how much diving do you want to do, what else do you plan to do beside dive and work, and are you looking for economical options?
If you are planning to travel in the high season (the Christmas holidays through Spring Break and Easter) the prices may be at their highest and places can book up quickly. This is true all over the Caribbean.
If you decide that you are interested in Grand Cayman you might take a look at Lighthouse Point, Coconut Bay Condos, or a 1-bedroom cottage at Cobalt Coast. These are places offering rental properties with kitchens, an onsite dive op with boat-diving, and a house reef for unlimited shore diving. If something appeals to you then try to work out a package for accommodations, shore diving, and boat diving. If you are planning to travel in late fall or winter then Cobalt Coast wouldn't be a good choice because the house reef often gets winded out at that time of year.
Here's another thought, you could spend a month in the Cayman islands but split your time between the 3 islands of Grand Cayman, Cayman Brac, and Little Cayman; they all offer great diving. There are more than 300 dive sites (boat and shore) across all 3 Cayman islands, so you would have lots of choices. The interisland flights only take a half-hour and there are a lot of flights into and out of Grand Cayman.
Good luck!