I will caveat this by saying I didn't read all pages before posting, but here is what I did/do for my 3 (yes, three) year old's tool's.
He and I have a standing "guy's trip to the tool store" every other weekend where he and I go to HF. They almost always have something on sale or freewith another purchase and so far he has:
--Small plastic toolbox
--3 measuring tapes (they are free most of the time, and we always seem to misplace one, lol)
--2 LED flashlights
--6 piece screwdriver set
--small claw hammer
--full (HF Cheepo) socket set
--some misc allen wrenchs that I had extras of
--my old 12v dewalt drill that I stopped useing after getting into 18v stuff
--misc drill and screw bits (yes I let my 3 year old use a drill to actually drill holes in things)
Between free items and sales I think we spent a whole $10 on everything, and it rerally makes him happy to have "his tools" when we are working on things, plus he stops "playing" with mine.
Beyond what my 3 year old has I would also include a few different size pliers (needle nose, channel locking, and etc.), maybe some wire strippers.
I know when I was 13, and also in a low income public assisted household, I was working summers in a gov assisted youth labor program and really knew how to handle tools. I think I got my first "Real" craftsman tools set when I was 14 or 15 and still have a lot of those tools.
So remember, there are 2 ways this could go, you can go cheep and give him a gift to use at 13 or go with a little more quality and get him a gift that he will treasure/use for years, but that is a gamble only you can make the decision on. For me I am going "cheep" for a 3 year old, but by the time he is 13 I would expect him to have much better tools, but I can also supervise him and know that he knows how to use and care for tools, you situation is different.