A .177 caliber air rifle wouldn't be humane imo, it won't reliably kill in one shot. The .22 cal air rifles will easily kill squirrels with head shots. There are a variety of operating mechanisms. Of them, the break-open models are the simplest to operate, so long as your son has the strength to operate it. The velocity doesn't really matter and speeds above 1000 fps harm accuracy.

Regardless of what you decide to buy, treat air rifles just like a firearm. Pellet guns can maim and kill people, so follow all gun safety rules.

Good luck hunting and report back

2015/16 Texas Outdoor Annual:
"pellet guns and other air guns are not legal for the take of any game bird or game animal other than squirrel. To be lawful for the take of squirrel, an air rifle must be designed to be fired from the shoulder and use the force of a spring, air, or other non-ignited compressed gas to expel a projectile of at least .177 caliber (4.5mm) at a minimum muzzle velocity of 600 feet per second."
https://tpwd.texas.gov/regulations/outdoor-annual/hunting/general-regulations/means-and-methods

I don't hunt with an air rifle, but it is a legal means. I use my air rifle for plinking and dispatching wounded critters in my suburban yard. My lab catches & plays with, but won't kill possums or squirrels....

A month of small game has passed. We went with a Benjamin Titan .22. My daughter had good accuracy (squirrel headshot) out to 30 yards or so.

Not much success so far but filling the freezer is not our goal. Most of the time we sit in the woods and listen to the squirrels laughing at us Still it has been fun getting into the woods.

We will try to get out a few more times before the snow falls. Maybe even try it throughout the winter.

I'll be taking my daughter deer hunting next week. She's 13 and hoping to take her first big buck this year, they are tough to find in the Big Thicket where we hunt. I harvested a nice old doe last week, so the rest of the season is focused on my daughter's hunting.

I hunt most small game with a 36cal flint frontstuffer.

but I keep looking at them. Maybe one of these days.

It is a calmer way of hunting, and you become a better shot, as you only have one.

Mine is the older version that only does 700 fps with a 14 gr pellet:

The newest version, the NP2, has a more powerful piston and does 800 fps with a 14 gr pellet:

I replaced the trigger in mine with an aftermarket one, which took all of 10 minutes and made the trigger simply amazing: http://charliedatuna.com/GRT-III%20Trigger%20New.htm

Mine fires 14.66 gr H&N pellets at 700 fps (http://www.pyramydair.com/s/p/H_N_Excite_Hammer_Pellets_22_Cal_14_66_Grains_Domed_250ct/1184), and is accurate and powerful enough to kill rabbits and squirrels at 40 yards. Plus, the suppressor it comes with works very well; it is the quietest gun I've ever shot.

The only downside is that the gun is heavy, so a slender 12 yr old may have issues carrying it for long periods of time.

On edit: I see you went with the Titan, which is the same gun that I have but in a wood stock Happy hunting!