My opinions, for what it's worth....
A first gun for your son to have as his very own? And one that he can use for years to come without worrying about buying up in recoil or caliber?
6.5 X 55 is an ideal choice. It's long-for-caliber bullets can create excellent penetration and it's recoil is well known for it's kindness. If you handload and have a modern strong action rifle you can create loads from 85 grain bullets up to 160 grains and from whisper easy-on-the-shoulder loads to harder-kicking elk stoppers with proper bullets and practice ... and bullet selection is decent. If you don't roll your own, or don't want to search for ammo, then a 7mm/08 or 308 or .243 would do just fine. Same with 30-30, 300 Savage, 7X57 .. it is really endless the # of good calibers for deer. Overall gun weight and fit is the real key to felt recoil. Along with proper technique and position when shooting.
A lot has been said about getting a gun small enough, short enough to be able to handle. Getting a gun that is real lightweight will only translate as a gun that kicks harder. A standard length gun with a slightly shortened stock is a great way to go. Get a bigger stock or longer pad as he grows.
It is easy to find light recoiling cartridges that are known to be good deer killers, but a gun that fits the kid well is worth far more.
There are so many cartridges that wil get the job done well, but finding the gun that fits you kid well is key.