Back in the Dark Ages when I was a full-time worker, my job involved having to handle large numbers of sometimes very irritable rats and rabbits. I was a total chicken with the rats, I used leather gauntlets (rigger's gloves). And because I have small hands, much smaller than my all-male co-workers, I had to develop my own handling techniques.
The gloves really were great. By wearing them they gave me a sense of confidence (perhaps misplaced) and this carried over into how I approached the animals.
If you approach an animal, even a rat or a hamster, with a tentative, darting motion, you will frighten the animal even more. It will be MORE likely to bite, or at least not respond happily. Try him with gloves - see if it helps him feel safer. Once he's able to touch gently and smoothly with gloves, get him to change food, water and bedding regularly and so get used to how the animal moves when his hand is in the cage.
I'm not accustomed to hamsters - I always thought they were just a variety (or another name for) guinea pig - and I'm (allegedly) a zoology major! But I do remember hand-feeding guinea pigs and baby rabbits - I really enjoyed it. Something to try him on - peel a carrot with a vegetable peeler, then hand-feed the strips of carrot to the animal. He can hold one end until the animal begins to bite on the other end and nibble away at it. His hand will be further away and safe. But do not try to handle even the tamest guinea pig or rabbit if your hands smell of fresh carrot or orange peel - I've had my fingers mistaken for food several times. Under those circumstances they don't do much damage but it startles you.
Lots of fresh vegetables is good for them, on top of the usual pellet food. I'm told you need to avoid feeding them iceberg lettuce, though - cos or mignonette is great.
So try the gloves, even if he insists on wearing the gloves when you don't think he needs to - they will help him gain confidence. I got to the stage where I could pick up rats easily with my bare hands. Not just grabbing the base of the tail, but put my hand around a rat's body and pick it up. That took a lot of practice and confidence. However, if I had to give them an injection (either general anaesthetic or hormone replacement), I always preferred to wear a glove on the hand holding the rat.
One of my co-workers (turns out, he was a sociopath) had taken home a baby rabbit at the same time I had. Same litter. Mine was a friendly, tame little thing who would run to meet me when I got home from work. His would attack him on sight, biting him hard. I could sit and pat mine, or feed him shredded carrot as a treat (where I learned to wash my hands, or feed him carefully). His would see a hand coming in close and lunge at the hand to bite, hard. His wife worked with us too, she didn't get attacked as much but still had trouble approaching this rabbit.
So he was going to give their rabbit to someone else at work. he brought her in and put her in a box in our lab. I watched him with her - he would approach her with an open hand then smack her hard across the muzzle, almost to daze her, so she wouldn't bite him (probably head spinning too much). I shooed him away and watched, and tried to work with her. I found she would attack harder at a hand coming in fast - that made sense, his hand WOULD come in fast when he was going to smack her. So I tried moving my hand towards her, very slowly indeed. At first she lunged, but at where she though my hand would be - only I was moving too slowly. So I got close enough to touch her and she let me. If she lunged at me while I was in range, I moved my hand away but in the direction of her tail, not back towards me (as I would if I were about to recoil to get a better hit at her). An approaching hand overhead - she would attack. Approach from the side, below her head level - less intimidating, she would allow a closer approach. Over a few hours I got her to the point where a slowly approaching hand from the side could actually stroke her and she would relax and enjoy it - rabbits LOVE to have the base of their ears scratched. Sometimes they like the side of their lower jaw rubbed but she couldn't come at that - too close to her teeth and old habits take a while to break. Similarly, under her throat was off limits - I suspect he'd also been choking her (*unmentionable Aussie swear word, referring to his parents' marital status*).
After questioning my co-worker further, it turned out he'd been too lazy to remove her from her hutch when it was time to clean it. He'd simply turned the hose on instead. "And it's not MY fault if she panicked and ran into the stream of the hose." Knowing this bloke, I'm certain he thought it was funny to play the hose on the rabbit. Then he opened the cage door to get out the food bowls - and she bit him. Don't blame her. I would've too.
By the end of this day the rabbit was responding to me well. He tried to pat her - as long as he moved his hand slowly, she would let him, but he didn't have the patience and really, was afraid of her (so much for being a big, macho bloke, supposedly experienced at handling animals).
He gave the rabbit to one of the secretaries. I stayed in touch, warned her that her new pet had been traumatised, and showed her what she could do and how to do it. That rabbit lived a happy, long life and was very loved. She eventually died of cancer when she was about six years old, her new owner very sad to lose her. My bunny lived until he was about 10, but got a bit cranky when he was older, hated to be picked up. He would happily hop over to me, though, and let me pat him.
If your son is timid with ANY of these pets, the animal will quickly become afraid of him, as happened with my co-worker (who should definitely have known better - he was a selfish, sociopathic idiot). That's why, if you're going down this path, you need to let your son wear gloves - something of heavy canvas or leather (not too thick). Even gardening gloves can help. Body language is really important - he needs to understand that the animal will be frightened or feel intimidated by certain actions. He needs to get into the animal's head. If he can, this is so absolutely marvellous for a kid to do. That's why pets are so good for kids, especially difficult children.
Often a puppy or kitten can actually be a better choice, if you're there to supervise your son while he gets the knowledge he needs. We're more used to carnivores as pets, we understand the way they play. Herbivores play very differently. They're also not as smart and therefore less adaptable. You can't trail a piece of string in front of a rabbit and expect it to pounce. But you CAN play tag with a rabbit - you have to try to touch the rabbit's tail with your toe, and not let the rabbit touch your foot with his nose. But do make sure you stop before the rabbit gets over-excited, or the bunny may bite purely out of getting worked up. We would play for a few minutes then stop for a cuddle.
Herbivores are generally not as smart, because they don't have to be. You don't have to sneak up on grass. This means it's more important to meet the animal in its space and not expect it to be able to adapt to you.
I hope you can get some answers. Hope this can help.
Marg