First of all, congratulations for wanting to finish your inherited GFG.
Would love to see a photo of the flimsy, now before you start working on it.
I'm sure others would too!
Like others, my thoughts would be to stitch/applique your hexie edges onto a border.
I'd keep it simple, and to just the one fabric, so as not to compete with the star of the show, your GFG!
As for what border fabric to use, I'd look for a tonal or blender that reads solid.
The tonal variations in those, usually work better with the old fabrics,
than a true solid does.
Sometimes using the back of a fabric as the front works well, when putting old fabrics and new ones together.
The secret .... audition! audition! audition!
Take your time to find the "right" fabric ... rather than saying "this will do", and having regrets!
Good Luck!!