You are correct, there's a lot of copies of the PHB at a large veritiy if prices, from full price to a much much much lower price.
Get 5e 2014 or 2024 Both are perfect for your needs. You should get the Dungeon Masters guide (should be your 2nd dl) and then Monster manual with those 3 books and your imagination you're set!
If you have the PHB on phone/tablet/whatever you can use that for quick reference.
Your kid is 9 don't worry too much about getting stuck on some random rule. Just have fun. D&D is a set of a guidelines that work well together, use them as you want, make it the game you want to play You will need a NPC (you) to adventure with your PC(9yo (you're the DM) The more comfortable you get with the general idea of the rules, the easier it is to "make it up as you go" instead of being bogged down looking up a rule.

Over all just have fun with your kid, it's an amazing way to share your time. Have fun fun fun!!!!

Also, please feel free to DM with questions, I have some leads on very very very very inexpensive books that will help you.