I have been using a 14 pound ball but have now developed tennis elbow. I have also struggled with wrist pain but it is mild. I have decided to drop down to a 12 pound ball, as I want to stay in the game. I realize that dropping to 12 will also very likely lower my average. I throw a mild hook (stroker style). When going down to a 12 pounder is there a recommended ball (make, model, etc.)? I wonder for those who use a 12 pound ball if there is a special make or model? I hear dropping down to 12 really changes the core of the ball, so I want to find the best one I can get.

Most bowling balls have similar specifications for weights between 16 and 14 lbs. When you get below that, most bowling ball companies offer a generic weight block plus filler for lower weight balls.
Also, I don't think there are many companies that advertise what the core looks like for various weight balls for a given model of ball; however, Storm Bowling does show what the weight block looks like for their balls.
In general, they'll show one type of weight block for 16 - 14 lbs.; however, for their highest level of balls, they do have the same type of weight block inside of the ball.
The difference between one 12 lb. ball or lighter and another is essentially the coverstock and not the weight block spec's, except for the highest level balls that they provide.
Talk with your local Pro Shop owner/operator and they'll help get you on the right track.
Take a week or two off from bowling and rest that arm. Once you feel like you're able to get back to the lanes, I suggest that you find yourself a coach to watch you bowl. You're likely using way too much effort to bowl. You're arm should be the pendulum for the ball rather than forcing anything. Use gravity more and you'll enjoy swinging that ball much more... Hope this helps...

Watch how much your ball will deflect as you drop from 14 to 12. It's amazing - my wife and my M-in-law have similar balls but my M-in-law's is a 12 (she's 80+ and can't hold on to anything heavier than 12) and the amount of deflection she gets compared to my wife is eye opening. . .

This isn't to say don't go to 12. . .

Back when I was using a 10-12 lbs ball, I had the problem of them breaking between the two finger holes,I was told that the plastic for the lighter ball is not as strong, true or not it stopped with a heaver ball.

Look into Motiv. They use the same cores for 12-16 pound ball. Below that they drop to a generic core. most companies go to generic at 12 pound but Motiv I notice usually says 12-16 on the real core.... I could be wrong on that but I would look into it. I used a 12 pound ball when I got back into bowling a little less than a year ago because of my Rheumatoid Arthritis and my bad back (Failed spinal fusion in the past). I used a Hammer when I was at 12 pounds and my average was pretty good relative to where I was as a bowler at the time. They seem to only use the real core. They just don't make balls at lower weights where they would have to change the core.