erase and reinstall OS X , mid 2010 mac mini.

When using Internet Recovery, did you choose "Disk Utility" from the macOS Utilities dialog and erase the hard disk before trying to install?

If macOS Sierra 10.12.4 or later was never installed on your Mac, macOS Recovery works differently:

• Command-R is still the recommended way to start up from macOS Recovery. This combination makes sure that the installation isn't associated with your Apple ID, which is important if you're selling or giving away your Mac.
• Option-Command-R installs the macOS that came with your Mac, or the closest version still available.
• Shift-Option-Command-R isn't available.

Have you tried creating a USB installer and trying the install from it?

How to create a bootable installer for macOS - Apple Support

Use Disk Utility on the installer and erase and reformat the actual

drive (not the Macintosh HD volume) with GUID partitioning and

macOS extended journaled. Then try installing macOS.

Just something to add as a aside (if you have not already done it).

Before you give away your Mini, you need to make sure all vestiges

of it have been removed from your Apple ID or any association to it.

What to do before you sell, give away, or trade in your Mac - Apple Support

Have you used Internet Recovery to erase your disk and install an OS?

Restart using one of the following methods:

Command+Option+R will install the most current version of operating system your Mac qualifies for (e.g. High Sierra)

Shift+Command+Option+R will install the version of operating system your Mac shipped with (e.g. OS X Lion).