Hey, welcome to ScubaBoard.
What sort of activities would your wife and son want to do when not diving?
San Pedro is by far the most built up for tourists so not surprisingly it's where most tourists go. It would have the most shopping, restaurants, bars, nightlife, etc.
If they're interested in some mainland activities, like Mayan ruins, cave tubing, jungle hiking, the Zoo, etc all of these are accessible from SP but they would spend a fair amount of time traveling back and forth by boat or air and car. In that case, you would be better off staying in Hopkins or Placencia. Both are on the mainland, although Placencia has a definite island vibe. I'd recommend
Hamanasi for the former. Not a house, obviously, but a super nice place with a PADI shop on site. If it's the latter, I'd recommend using
Splash Dive Center. You should be able to rent a house there easily enough. Patty Ramirez, Splash co-owner and operator, can help with the accommodations.
By "wide range of dive sites" I suspect you may be planning to do the Blue Hole. There's a thread a few threads down that discusses the BH at length and you probably should read it.
It sounds like this is your first time to Belize. You'll find a wealth of information in the threads here. I also recommend
Belize Insight Travel Guide. It's not a dive guide, although it will mention some sites, but it's a really great travel guide that breaks down the country district by district and has a ton of glossy photographs.