Would a potion of fortify destruction increase the damage on destruction-based staves?

I don't think so. I believe the staff has a set damage.

Yes, potions of +Destruction damage do affect your Destruction staves.

Test case:
Falmer Staff of Chain Lightning vs. Nazeem.

Without potion: 75 -> 34.98: 40 damage (expected.)
With 50% potion: -> 14.98: 60 damage (proving a boost.)

Great, thanks. I had read that certain perks in some of the magical skill trees would boost staves, but I couldn't find anything about potions doing it. thanks for testing!

Fortify Destruction Potions do increase staves damage output -not sure about the named ones though.

The Staff of Jyrik Gauldurson for example doesn't seem to be affected -propably because it uses a unique spell, not available to the player otherwise, but through the staff itself. I'm guessing the Staff of Magnus won't be affected either, but I haven't done any tests to know for sure.

Wow, I might have to make a new Gandalf-type character who is good at Alchemy and just uses various staffs (staves?). Did I read once that there is a way to make your own staff in Dragonborn?