Put me in the "make it special for this character" camp.
I like the idea of it being some kind of special weapon of Elven design. Mithril+magic allows the blade to be unnaturally light; but a blade that light requires substantial, dedicated training in order to use as effectively as a "normal" greatsword, so it's an "exotic" weapon proficiency that must be taught by a master. Even better would be if it has some minor-but-nifty abilities, such as:
- Absorption. This special greatsword was designed to be sympathetic to external magic fields. For someone proficient with, and attuned to, a specific blade of this kind, it is possible to invoke a ritual that captures part of the magic of a different weapon. No special properties can be transferred, as they cannot be worked into the blade, but if the owner finds a normal greatsword of a higher enhancement, she can spend an hour to conduct an arcane ritual that will transfer that enhancement bonus to this sword. The other greatsword is destroyed in the process, its strength exhausted to empower the mythril greatsword. The ritual will fail, causing no change, if the other weapon is not a greatsword, or if its enhancement bonus is equal to or lower than the current enhancement bonus of this weapon.
(With this property, the character no longer needs to worry about abandoning her old sword--but also can't just take any old +1 dagger or whatever and soup up her greatsword.)
- Glowing. (Everybody loves glowing/harmlessly flaming swords.) Perhaps it glows in the presence of particular enemies, or when near running water or a tree that was recently cut down or something.
- Singing. The potent underlying enchantment of this greatsword causes it to constantly produce small sounds as long as it is outside of its scabbard. When at rest, it hums quietly, and a soft tap to the blade produces a bell-like ring which seems to linger longer than it should. When swung in combat, snippets of airy, tuneless music can be heard; whether they are influenced by the current course of a given battle is hotly debated.
The first is meant as a quality-of-life kind of thing, just making things easier on your Ranger friend--making it perfectly reasonable that she'd keep one signature weapon for her whole career. The other two are just examples of neat purely-cosmetic quirks that seemed appropriate for a beautiful, magical, elven weapon.