My PC has over 200 GB of free storage but iTunes won't back up my iPhone with 64 GB

Usually when I see this issue, it's one of the following:

1. The iTunes backup folder is missing, damaged, or has incorrect permissions.
2. The iTunes backup folder was previously redirected to an external drive that's now disconnected (a fancy variant on missing really :))

There's two locations for the iTunes backup folder, depending on whether you installed iTunes direct from Apple or via the Microsoft Windows 10 store.

Direct from Apple:

1. In the Windows search bar type %appdata% (with the percents) and press return/enter. A file browser window will open to the Application Data folder.
2. From there, navigate to Apple Computer -> MobileSync -> Backup.

From the Windows 10 store: 1. In the Windows search bar paste %userprofile%\Apple\MobileSync\Backup and press return/enter. A file browser window will open to the iTunes backups folder.

Do you have one or both of these folders? I usually like to start by recommending renaming the existing Backup folder and making a fresh new folder named Backup (must be named exactly Backup) and seeing if that helps.

My blog post about how to find the iTunes backup folders: https://deciphertools.com/blog/where-are-iphone-backups/ (Obvious disclaimer: I'm linking to myself as a reference.)

I have more suggestions about fixing an error making a backup here https://deciphertools.com/blog/fix-iphone-backup-corrupt/ under the section "iPhone Backup Corrupt or Not Compatible - Can't Back Up" There's tips about errors restoring a backup as well, but please don't try those for this issue!

Here's a fun corollary to Kelly's answer:

The error can also appear if your iOS device doesn't have enough free space.

Yes, the error's wording is completely wrong in this case. It says it's your computer that's lacking space, but, if you check Settings >> General >> iPhone (or iPad) Storage and it's almost full, it could be your iOS device.

Just had it happen again... I followed the accepted answer, still no love, even tried installing iTunes as admin and running it as admin by right clicking and saying, well, run as admin, no dice. Tried on the D:\ drive, even though both C:\ and D:\ have over 150 gigs free and it's a 32 gig iPad. I get the "not enough free space on your computer" error complete with unhelpful "emptying the Recycle Bin" message.

Vaguely recalled this happening before, deleted a few gigs of stuff from the iPad, and now all is happy. I just completed an encrypted local backup.

Again, the error said my computer didn't have enough hard drive space & said to empty the recycle bin, etc. But it was, indeed, the iOS device that needed free space for the backup to execute.