> I meet someone named Jane Doe, and then while trying to think of a mnemonic to remember them with, something else happens and I lose the moment.

It's a good idea to spend some time pre-calculating mnemonic images for names before you are in a social situation where you need to use them.

Find lists of the most common first names and surnames and prepare against random combinations of those.

This process will also help you get better at doing it for the less common names on the fly.