In that price range the only real jump to make from a Ruger would be a S&W Model 41 or a nice, used High Standard like the Victor, Citation, Trophy, etc.
The Ruger, S&W, and HS are all guns that will give a lifetime of service without wearing out, so I wouldn't worry about that.
If your son is going to stick with iron sights, then a longer barreled gun will give him an advantage of increased sight radius. I don't think I'd go with a 10 inch barrel though. Not when the shooting must be done with one hand. Six inches should do it.
If he is going to mount a red dot sight, then a shorter barrel of about 4 inches would be better to save weight.
Since he is still starting out in this sport, I'd stick with the Ruger for awhile longer. I'd upgrade the trigger and sear with Volquartsen parts at a minimum. These parts you should be able to install at your kitchen table, no problem.
What are his scores like with the stock Ruger Mk III?
A couple of years ago I read an article about the Rugers in the NRA's competition magazine.
Basically, you shoot the Ruger stock until the scores plateau. Then you make an upgrade. Say, the trigger and sear. Then you observe the scores for improvement. When they plateau again, you add a set of custom grips. I like the Herrett stocks with the palm shelf. The Volquartsen volthanes are pretty good too. And there are others, probably better. Look at NealJGuns.com or Champion's Choice.
Once again, scores will improve but then plateau. The next step would be to transition from iron sights to a red dot. In case you are not sure what I mean, I'm talking about a scope that projects a red or green dot onto the front lense. Not a laser sight that projects a red or green dot onto the target itself. Don't mean to offend if you already knew the difference.
Ultradots and Matchdots are considered the best for bullseye, but I like the Millet sights just fine. A 3 moa size dot should work.
Now, after his scores plateau from that, and he should've fired many, many rounds by this time, you might consider the S&W or the High Standard.
Or you could take it to the next level and get a Pardini or other somewhat pricey European .22 like the Hammerli or Finewerkbau. At this point the only thing that could possibly be holding him back would be the quality of the trigger pull. The VQ trigger and sear will vastly improve the Ruger, but it will not put it in the realm of the High Standard or the European match pistols. I offer no opinions on the Model 41's trigger pull, as I have no experience with it. I've owned various High Standards and currently shoot the Pardini.
But by then there will be other things to consider. By this time he'll likely be wanting to shoot the .45 and centerfire events of bullseye with the big boys, not just the .22 stuff.
Instead of another .22 (he'll be able to shoot that Ruger one-handed with almost scary precision by this point anyway), think about getting him an M1911A1 bullseye wadcutter gun.
