Our family has a Macbook for regular computing, but my sons enjoy playing a game that's not available on Macs and has the following minimum and recommended system requirements to play:

Operating System
Windows 7 32-bit (MINIMUM); Windows 7 64-bit (RECOMMENDED)

Processor
Pentium 4 class machine minimum 2 GHz (MINIMUM); Intel Core i Series or equivalent (RECOMMENDED)

Memory
1 GB (32-bit OS) 2 GB (64-bit OS) (MINIMUM); 4 GB (RECOMMENDED)

Video Card
128MB video RAM (MINIMUM); Hardware Accelerated SPU - 512MB video RAM (RECOMMENDED)

Resolution
1024x768 (MINIMUM); 1280x1024 or higher (RECOMMENDED)

Disk Space
650 MB free (MINIMUM); 1 GB free (RECOMMENDED)

I'm no expert, but it sounds like these are not requirements which would translate to an expensive laptop. The issue is I wonder if I should buy something more robust in case they want to do other, more complicated gaming on that PC. If so, what would the cost delta be?

Long story short, I don't want to pay for more than I need, but at the same time don't want to cheap out and get something which is a paperweight except when it comes to one game.

Any thoughts or advice would be greatly appreciated.

How old is your son and how generally in to gaming on PC is he?
Are you ok with a desktop for this use instead of a laptop? You still get more perf/$ if you have the space to set up a desktop.
This one on sale at Newegg would take on basically any general use and gaming up to 1080P and I believe even meet the requirements for a VR headset.
https://www.newegg.com/Product/Produ...61&ignorebbr=1
Then add on a 23-24" 1920x1080 monitor for between $100-150, and a pair of headphones for <$25
Dell's Inspiron Gaming desktops look like pretty solid value for 1080P gaming desktops as well.
http://www.dell.com/en-us/shop/dell-...gaming-desktop

That would get you a solid gaming system that is also usable for basically any general purpose tasks.

If you'd like to spend <$500 all-in, I think you'll need to look at refurb units a generation or two old, but it should still be doable.