We spent a week in Roatan last May. On the West End you can find lodging as cheap as $20-30 nt. or less. A/C usually costs $10 more. Dives at most of the shops in the area are $25. I recommend Coconut Tree Divers in that area. It looks like it's about $300 for the course through them.
Coconut Tree Divers - PADI 5-Star IDC Dive Resort on Roatan, Honduras - scuba diving courses, divemaster internships, instructor development, tec diving, and more! Roatan would also be a good place for beginning dives, there's good stuff to see in under 30' of water. One of our best dives after going deep off the wall was drifting under the boat in 20' of water, all of sudden there were turtles everywhere.
Surprisingly several in our group commented on the high price of food. I didn't think it was any more than anywhere else in the Caribbean though. Everything costs more on an island. A good site for Roatan options is this one:
Dive Bay Islands
I just received an e-mail from Delta, they just started $599 r/t flight to Bonaire from JFK. Saturday only. You could stay at Golden Reef Inn or Wannadive very reasonably. And get certified through Wannadive. Most of the resorts on Bonaire offer a package deal of accommodations, diving, truck rental.
Info Bonaire - The Bonaire Information Site - The Most Current and Up-To-Date Information has a lot of detailed info. Bonaire can be a little more expensive though for food - everything is brought in from Curacao or elsewhere.
If you're by yourself, Bonaire is an easy place to meet/dive with other divers. Most people rent 4-person trucks - it's the default dive vehicle there - so have room for others for shorediving. And if you hang around any of the dive resorts, local dive bars etc. you'll likely be asked to go diving with someone - I was at least 4 or 5 times while waiting for my buddy etc. One thing about Bonaire, the diving is so good that it will likely ruin other places for you - it's consistently rated the best diving in the Caribbean in a lot of categories - except no big pelagics. (Sharks, etc.)
Personally I wouldn't go to Jamaica to dive. It's not the greatest. And Cozumel isn't a place for new divers necessarily, a lot of sites are deep, drift diving. Better enjoyed when you've done a couple dozen dives first.
One point to consider wherever you go is that you may want to do the classroom part of your training at home so that you don't have to waste vacation time doing it. You would get a referral from your instructor to be completed by the instructor you dove with on your cert. dives.
PADI offers an e-learning program also so you can do that part at your convenience. They're going to be the predominant dive organization in the areas you named.