What a great question! That term is bandied around all the time, but I too don't think I've ever read of a concise, generally agreed upon definition of what it means.

I guess I have come up with my own understanding of what it means for me, which I can share. When I say "I have processed a memory" I generally mean this: I have worked through the dissociated elements and taken 'ownership' of it. So for me a 'processed' memory is one in which I own the narrative and accept it happened to me. I no longer have intrusive images or flashbacks about that particular memory and I have hold all the previously dissociated aspects of it in one place.
I know when I have fully 'processed' a memory because it no longer causes me any distress and there is nothing about it that causes me to dissociate any longer.... I can talk about it easily as an aspect of my own personal history, just like any other.

That is what a 'processed' memory looks like to me.

The ones I still need to work on ('process') are the ones that still cause me to dissociate.