Welcome aboad Nancy.
A question if you dont mind- Do yall where you live have to hunt deer with a shotgun? If not i personaly wouldnt unless forced to by law, regs& conditionsetc.

Nothing wrong with those 12 guage shotguns mentioned - It was also my first gun at 10& what i used for deer first few times too.(it was the law)

Many firearms will come in youth models no sawing off of the stock needed to fit& you can get a fullsize stock later if need be .( there ussallyjust a little shorter a inch or so-so they do fit adults too with shorter arms etc also)

If i was buying for my son where i live his first gun think id go for one of those Nef single shotrifles in 270. 280. or 25.06 & i would get a 12gashotgun barrel& a 22. barreletc that would give him 3 guns or more on one frame.( the barrels are interchangable/switchable)

They can be had at walmart at very reasonable prices( about $200.00) with the other barrels extra cost add ons/also reasonable in price)

If ya do a net search for nef rifles or h&r rifles you can check them out on there websight& see what the extras cost etc too.

Good luck , im sure he will injoy whatever you decide on.