Weapon Refining

I would go for the dragon upgrade, because it imho brings the largest amount of value - you get increased might, hp and a stat increase depending on your preference, all that combined with the reworked dmg calculation which will let her destroy high res low def mages, which she can now afford to duek safely due to her own high res.

As for the legendary weapons (out of those you mentioned) I think that Eldigan's upgrade is the best. You'll probably go for the fury upgrade which will allow you to choose a different A-slot which is incredibly valuable. Linde's upgrade is also good, but only if you don't have Delthea and if you do, you should check and compare their IVs because my Delthea actually has better statline than my fury Linde, so I would never do the upgrade in my specific case. Julia's upgrade just doesn't bring enough value for the large investment imho, so I would place it lower on the priority list.

Eldigan gets double fury which is a monster upgrade for him.
Julia is getting the dragon killer which is amazing, and linde gets amazing team buff.
so they are all good, choose whoever you use most.
I go for Linde first and Eldigan later.

In terms of Legendary Weapons I am most excited for the Hautclere upgrade!
Just imagine a Michalis with Ignis +10 damage! Or Minerva with an offensive build!
I'm hyped.