It was fitting for it to have one. Magical weapons have names in our world. It's much easier to say "IcingDeath" rather than "the frostband weapon with the black adamantite handle and silver blade with diamond edge that was discovered in the lair of the now slain dragon, IcingDeath". It's a unique weapon, hence, it should have a unique name. Some weapons even are sentient, although IcingDeath is not.
You wouldn't give a name to a normal, magicless item or an item with only a common set of enchantments, of course. That would just be silly. Even so, some still name an item if its shape is uniquely crafted... or if they're simply so full of themselves that they think their weapon deserves a name when it rightfully does not.
For example, my other Scimitar is named Twinkle because it has unique properties. However, my armor and bracers are not unique, even if they have magical properties, so they are nameless.
[TL;DR: blah blah blah unique weapons have names blah blah his other scimitar is named Twinkle.]
Drizzt sounds like a D&D nerd right now. I'm dying.
You wouldn't give a name to a normal, magicless item or an item with only a common set of enchantments, of course. That would just be silly. Even so, some still name an item if its shape is uniquely crafted... or if they're simply so full of themselves that they think their weapon deserves a name when it rightfully does not.
For example, my other Scimitar is named Twinkle because it has unique properties. However, my armor and bracers are not unique, even if they have magical properties, so they are nameless.
[TL;DR: blah blah blah unique weapons have names blah blah his other scimitar is named Twinkle.]