This is somewhat, at least psychologically, similar to where I'm at with my wife right now:
When we first met, she'd turn green at the thought of eating venison. I never pushed it on her, but made it available, should she want to try it. While visiting my sister and brother-in-law in NC (he was stationed there while in the Marine Corp), my brother-in-law made biscuits and gravy with venison sausage, as well as frying up some back strap. My wife tried it, liked it, but mentally forced herself to NOT like it. I left it alone, and didn't say much to her about it.
Her uncle's brother-in-law hunts, makes has summer sausage and slim-jims made out of venison. My wife started eating that because it tasted good, and she started ignoring the fact that it was venison.
Move on to two years ago, I get a little button buck on opening day. I took it to her cousin's where we processed it. She didn't help with the butchering outside, but she did help package it (she wouldn't touch it, but she doesn't like touching any type of raw meat).
Last year, while my sister and brother-in-law were up visiting for the 4th of July, I decided to get my bb-gun and .22 out to do some plinking. Boy, my wife took right to that "little" .22. To the point where I may buy her her own one of these days.
My wife now goes with me when I hunt at my parents' cabin (she loves to sit inside by the fire and read a book), and she keeps asking me if she could hunt turkeys with a .22. She's also asked about taking a hunter safety course so she could get her license, should she want to go that far.
All of this happened over the course of about 4 years. I never pushed her on any of it, but made the opportunities available to her. The same could be true with your son.