How to wipe an old white MacBook? Reading over Apple's documents on how to wipe a machine, it's all about internet installs. Now it's been so long since I did anything with this machine, I don't recall the process... was 10.7.5 an internet-based install?

What options do I have to wipe an old white MacBook?
The short answer is...

Using OS X boot media or your recovery partition (if available; Command+R), boot into it like you are going to do an install.  However, when you get to the first screen, launch Disk Utility and do a "Secure Erase" of your hard drive.

Once that is done, you can stop there and have a white MB with no OS or you can continue and install an OS for the next person who becomes the new owner.

Can your white MB support Internet Recovery?

If it is the 2010 model (their last one before resurrecting the line last year) and you had Lion installed with the EFI Firmware update, you can use Internet Recovery.  https://support.apple.com/en-us/HT202313

There's no harm in checking...just press Command+Option+R during boot.  If you get to Internet Recovery, then you have it.  If not, then chances are, your MB doesn't support it.