Recommend me a sub

I used to have the same car. A single 12 ported does take up a good amount of space. I had a box about 15x15x28 I believe and even though I still had enough room for my needs, IDK how much your nephew is gonna need for his daily purposes. A 10 seems more practical but I would get to measuring and figure out how much he's gonna really need. Or alternatively you can seal off the cabin and trunk into a blowthrough setup with sub facing forward, which is more efficient actually and allow more trunk space as well. An infinite baffle setup is also a great space saver too, look into it. The Dayton HO is always my go to sub when it comes to small boxes and it's less than $200. It's more SQ oriented so if he wants big bass and don't mind a bigger box, there are many other great contenders for the price. 

You'll need to upgrade the electrical system in that car or run the amp at a higher ohm load like 2 or 4 ohms.

I had the exact corolla. It has a really high vehicle peak frequency. If he likes rock and edm, he can just keep it like that with a moderately high tuned ported box 35-38hz.  if he like rap or anything with deep bass, he'll need to deaden his trunk because the car is flimsy as fk and has a hard time hitting lows naturally without acoustical treatments.  No blowthrough, there's too much air holes and pockets in the rear deck, your output will be sh*t and phase gets fked unless you do some fiberglass and seal off the rear deck completely off from the trunk.  

Noticed literally zero output gains from the big 3 upgrade or alternator. The biggest gains in output I had was when i added batteries to the rear next to the amp, that was complete night and day difference in output.  

I'd go with a shok industries triton 12 d4, run it at 2 ohms then do two 12s later on when he doesnt need to give up as much room. Those were the loudest cheap subs I've had on that car.