Where is his Ostomy located? It sounds like it's above his belt line? Does his stomach pooch out? It's harder for men I think, especially if they have any gut at all, because above the belt line shows it all, below there's not much room. Women have many more clothes options.
I don't know that a belt will help him, the pouches usually hang down through a hole in the belt. They're used more to keep the flange or adhesive part on if you have a difficult/unusual stoma, not so much to stabilize the pouch itself. Does he use a 1 piece or 2 piece? If he's having issues keeping it on while active, a better gift would be a visit from a good ostomy nurse (ET nurse). If you know a belt will help, I'd go with NuHope, they make the best and have wonderful customer service.
If it's an issue of it showing, if anyone's handy with a sewing machine, you can make him a custom fit "cumberbun" or a stretchy type of tube top that he can put on (maybe with velcro) to keep it close to his body.
As he heals and feels better, there should be no reason he's not just as active as he was before the surgery. it shouldn't interfere with his life at all, as long as he can get his appliances to fit and stay on properly. Stomas are pretty sturdy, and not easily harmed, he shouldn't worry about
that, and a good nurse can reassure him of that fact.
As for odor control, my colostomy had way more odor than my ileostomy, but the thing that worked best for me was always the M9 drops. They're available at most medical supply stores and should be covered by his insurance. Just make sure he uses way more than it says to use, a couple drops don't do a thing for thick output, you need a good squirt or 2, enough to really cover it.
You may also want to buy him some nice shirts for Christmas that are made to be worn outside the pants, not tucked in. Does he golf? Tommy Bahama makes some great shirts like that. But if you look around, you can probably find something.
You're a good child (son/daughter, sorry can't tell if your male or female from your signature) to be so thoughtful of what he's going through. This surgery is a huge body image adjustment for most of us, even when it was elective. Many of us went throught some hard times afterwards, a little depression should be expected, and it may take some time for him to adjust.
I hope they can get his cancer in remission. Keep us posted.