Nephew turns 13 in 2 weeks and as the cool uncle I'm gonna give him a rifle. Not sure which one yet so help me open my eyes.

10/22 of course is always on the list.
Ruger american rimfire

or?

Also thought about a 22 ar15 but did they fix the issues with them?

I like my 10/22 our you could look into a bolt .22. The ruger ones look nice.

10/22 if he already has some firearms experience, or if you don’t mind keeping him stocked with a few bricks of 22lr. Plus, infinite upgrade potential.

The American Rimfires are nice bolt actions, same ballpark price as a 10/22.

Either way, you will be the cool uncle.

10/22, because then you can help him customize it to celebrate future birthdays.

This is the final product in my build for my daughter, when she was younger...and it only weighs six pounds, including the scope:

A lever action or pump 22 would be fun as well.

22's are something I really appreciate more than a lot of gun guys so I have a number of them to suggest.

#1 The 10/22 as you mentioned. It will cost you anywhere from $199 to $250 for the base model (typically I find them around $220 where I live). Endless parts and upgrades. Just a little more than some of what else is out there.
#2 Marlin Model 60. It comes in about $179 I have had 2 of these things and they are the most reliable .22 I have shot. It doesn't complain about any ammo. No hiccups in either rifle if I remember right. (Keep them clean of course)
#3 Savavage Model 62/64. Reliable as well. Very limited after market parts and upgrades. I don't really thing there is much of anything out there for it. It only cost's $99 at your local Wal-Mart though! Thats the best part.
#4 Chiappa Lever Action Rifle. It takes down. I don't remember the exact model number but it is a ton of fun. The iron sights suck but to through an optic on it would be kind of silly to in my opinion. Its really just a fun 50 yards and in gun. It runs about $250 I think?
#5 Savage A22. I don't know if there is a better looking stock gun. They are just nice and good looking. Friend had one and I shot about 25 rounds through it. Its a nice gun. I like them. Don't know the cost.

I played this game a few times over the years from the cool older cousin to the fun uncle.

First, I would take them to a local dealer w/range and rentals. Let them try out a few different things to see what they liked and give pointers and suggestions.

Second, use that intel to either get them something they would like (this could be based more on looks than anything at this age) or take them (with legal guardian to fill out 4473) on a shopping trip to let them pick it out.

At the end of the day, anything you get would be seen as "cool uncle" status but IME if they picked it out it will mean more to them and that status lasts much much longer.

S&W's 15-22 are really decent for the price. They are reliable and mags are inexpensive. They also work exactly like an AR-15.

There is three, I see at the local steel match regularly. They all run very reliable, better than the 10/22's. I have a cheap $25 Tasco red dot on my 15-22 and can nearly win most matches with it.

I got my son a Ruger American Rimfire when it first came out.  For his upcoming 12th birthday, a 10/22 since it can use the same mags as the RAR.

RARc

Why start a young man out with something that has a shit trigger like a 10-22?

Everyone in here is assuming a rifle and thats fine. Excellent suggestions so far.

Open to the possibility of a .22LR handgun?  Lots of great options. A Ruger Mark of some sorts would be ideal. Or a Buckmark might also be a good option. Most popular handguns, Beretta 92, Smith M&P etc... also have a .22 version clone of their centerfire versions. There are also plenty of .22 chambered revolvers out there.  I love my Ruger Bearcat single action. Always tags along during range trips. Just throwing the .22 handgun idea out there.  For a rifle can't beat the classic 10/22.  I had an old Winchester model 190 when I was a kid.  We shot the snot out of that thing and killed many bottles and cans with it.  Kinda hard to find these days but I see em every now and then and I have toyed with the idea of picking one up again.