Depends on your type of stainless
The sort of stainless used for guns tends to be pretty specific. The other alloys that are highly stain resistant, like the stuff used on boats, does not have the right sort of mechanical properties to work well in firearms. So for the cost to strength and wear issues we're left with guns using the sort of stainless alloys which are only rust resistant. Not rust proof.
Smcx, since there's no stainless versions you're left with the more expensive options. Marlin 39, Winchester 9422 and Browning BL22. But any of these will cost a lot more than the alloy receiver Henry.
I'll suggest that the Henry is a lot tougher and durable than you think. It's also about half or less than half of the cost of those other all steel options.
My own basic model Henry lever has something over 5000 rounds through it at this point. When I took it apart to detail clean it a few months back due to a sense of guilt I found two things. First was that I didn't need to bother cleaning it. The only stains it had were easily reached through the ejection port. The second thing I found was that the "cheap alloy" parts were doing just fine. The ONLY sign of wear was a slight shininess on the parts that rubbed against each other and the paint on the bolt wearing off where it runs in the rails.
It's a first class shooter that is very accurate and from all accounts of folks that have owned them for many years and over 20K rounds downrange and still going strong that I'll be passing it on to my descendants still in fine operating order.
By all means spend more for all steel if you like. But they won't shoot any better.
I take mine along to my club's Junior .22 night to let the kids shoot. Which is part of the reason why it's up well over 5000 shots yet only about 4 years old. The 10 to 15 year old kids love it because it's light and great fun to cycle the lever for each shot. The lightness comes from the skinny round barrel and alloy receiver. The lighter weight is something you'll want to keep in mind for your own boy. Regardless of which brand you get I'd suggest a round barrel over a heavier octagonal style.