Eclid, Bow of the Hunt: Hunting Bow: Silent Moons + Huntsman's Prowess
Jali'agul (He-Who-Endures): Dwarven Short Sword: Adsorb Health + Absorb Stamina
Phage & Tyne: Twin Ebony Daggers: Shock Damage + Absorb Magicka
Alarana, Song of Morning: Silver Sword: Fiery Soul Trap + Fire Damage
Tharinholt: Warhammer: Notched Pick Axe + Shock Damage
Atm I have a Nord Hero Sword that I kept unenchanted so I can use elemental fury.
However I always carry a Stalhrim sword enchanted with frost damage + shock damage that I named "Dragon of The North". I believe it deals around 190 physical damage plus 92 frost damage and 60 shock damage (I have the augment perks).
It's pretty op but I only use it when I REALLY want something dead (I'm looking at you Legendary Dragons)
What type of name are you looking for? Something descriptive (i.e., Martin)? Fantastic (i.e., Tolkein)? Pragmatic (i.e., TES generic weapons)? Here are a few quick ideas to cover each of those conventions.
I think of clever and sometimes funny names for my enchanted weapons. For example, any bow with a frost enchantment I call "Jack Frost's Bow". Or any weapon I made at the skyforge when I enchant it I have skyforge at the beginning, for example Skyforge Dragonbone Greatsword.