Can't go wrong with 7mm-08 for caliber.

1st Question 1st - What kinda budget do you have to spend & are YOU gonna want to use it until it's time to start teaching Small Son??

Then we can start on the ...What to buy for a Rifle for a small sized preteen beginning shooter & capable of lasting a lifetime if need be.

IMO the criteria for a rifle should be: Weight 'tween 5 & 6lbs, in a youth sized short stock, or an Adult sized throw away stock trimmed to Youth Length, with adult sized replacement stock available, then pick a Little Red Riding Hood do all caliber with not too much recoil, but useful forever. That sound like your perceived Outline of Need??

Several ways IMO to go either in a Bolt Action or Lever Action, but a Single shot?? ehhh!! not so much, but doable ...but more than likely gonna become a Kleenex throwaway gun.

In bolt actions -Budgets not withstanding - 'tween 5 & 6lbs are:
My 1st OverAll choice is a Tikka T3's blue/syn & mebbe a used SS/syn,
then mebbe a
Savage LightWeights, Savage Youth series,
M70 Short Action Pushfeeds on 20" barrel Compacts,
Ruger UltraLight in a SA caliber,
Kimber Montana @ 5bl 2ozs in SA calibers.
Zastava Model 85 aka Mini Mauser, also sold under 3-5 other brand names from Remington, Charles Daly to Mark X and an ideal sub sized small receiver rifle in several useful calibers but the 7.62 x39 is what I'd buy as it is equal to or better in performance than a 30-30 in name brand ammo & imported "training ammo" is super cheap,
CZ 527, another ideal sub sized receiver in 7.62x39 or other calibers @ 5 3/4 lbs w/ a 18.5" barrel.

Lever Actions: you can never go wrong with a 30-30 in this kinda circumstance. Finding a quality gun is the issue now that Marlin has been bought out by Remington & apparently the QC has gone the way of the DoDo bird. Find an old Marlin in good shape and you could call it good. Dunno nuthin about the Rossi's but it'd be a possibilty if you are determined to buy a lever gun.

Calibers: Since we are looking at low recoil, pre teen child tolerable but WTail & Hog but Lifetime useful ....I'd suggest several different calibers besides the perennial kneejerk response of Ya gotta get a 243 or 223 for a kid. Those 2 calibers will work but I believe there not only Other Options but potentially Better Options. As Said earlier, your 6mm Mohawks ammo is about dried up & all gone for lack of sales volume.

Under no circumtances would I buy a RUGER 16" barreled Compact as the muzzle blast & concussion cone is simply fearsome awesome to a preteen shooter.

Cutting and re crowning a Used longer than needed barrel is not horrible spendy - a guy here on THF offers it $50 as I remember - to do and does not Affect accuracy, but may actually improve it, or Effect MV past about 35-50fps at most per inch in barrel length reduction .... until you get below 20"s. Short arms handle short overall rifle lengths waaay Mo Bettah than longer barreled guns was my experience with my 2 sons at that age.

Tikka T3 synthetic factory stocks can be cut down easily and used synthetic factory takeoffs are common from folks @ $100 or less upgrading into a B&C Medallion, M70 Fwt New Haven stocks can still be had from CDNN in Abilene for $50 in 100% finished uncheckered Fwt stock or $80-$100 for checkered Fwt brand New stocks leftover from the Plant closure, Ruger UL receiver is a drop in on aftermarket less expensive stocks, but I'd leave the full length factory stock untouched.

2nd things 2nd - Calibers: Without stepping on the usual toes and exepecting the typical kneejerk reaction for a pre teen childs 1st rifle of "Ya gotta get a Kid a 243" or "223's will kill any deer in Texas just like a ...you name a caliber"....I believe that not only are there OTHER choices, but potentially BETTER choices especially for a Lifetime Functionality. I too would focus initially on recoil like the 243 & 223 guys are correct about....but Suggest thes calibers in the same recoil range:
#1 a 6.5x55 ( Tikka T3 Hunter Grade @ 6lbs 10oz in a blue steel & wood stock or 6lbs 2ozs in a takeoff factory synthetic stock),
#2 a 7.62x39 in either a M85 Mini Mauser or CZ 527 Carbine,
#3 a 257Roberts,
#4 a 260 probably available best from Savage or uggh in a Remmy 700,
#5 a 7x57 Mauser hardest to find and heaviest,
&
#7 a 7-08 ( again in a Tikka T3 synthetic as my best choice)
...and study the pro's and cons of these for your requirements with an open mind.

My Opinions are mine and based on MY Experiences with MY family and the 13 years I spent of calling on several hundred Gun Dealers in 5 SWestern States...and Your Mileage May Vary...but that's OK too.
Have Fun
Ron