I'm working on a custom class, but can't seem to think of a name for it.

Specialization: Combat

Major Attributes: Intelligence, Agility

Major Skills: Long Blade, Block, Medium Armor, Destruction, Security

Minor Skills: Light Armor, Short Blade, Mercantile, Athletics, Restoration

The character is a Dunmer, if that helps.

A sort of do-it-all character? A class name that I've been using lately is either Tradesjack (for male) or Tradesjane (for female.) It's a shortening of the phrase, "jack of all trades." Most of my characters are the sort of do-it-all types and the typical "ranger" or "hunter" or whatever type of name doesn't fit with their RP. I use this as a nice easy way to get naming my class out of the way with a minimal amount of thought.

How are you playing your character? That could mean as much as the skills. It looks to me like you are playing as a Dunmer version of a Rogue.

Combat, Agility, and Intelligence with a hint of magical influence because of your dark elven nature.

A biography helps here.
Right now my char is a male Imperial an old Legion veteran, who kicked around the emprire for years getting in get-rich-quick schemes with the Thieves Guild.
He did a tour in Vvardenfell commanding a company of Scouts. Since the Thieves Guld in Morrowind wasn't strong he naively got mixed up in a smuggliing scheme with the Cammona Tong who set him up for the bust.
He was kicked out of the Legion and got a life sentence in the Imperial hoosegow. (There's more to his story but...)
The obvious name for his class to me is ex-Centurion.

As has been said, if you really want to focus on role-playing (And since you're putting enough thought into a name to ask about it here, I'd assume you are.) you should probably take into account how you're going to be playing the character. After all, while in Morrowind, the default classes are just specific skillsets, in reality, a thief is a thief because he steals things, not because he happens to be good at sneaking.

Often, though, when making custom classes, I just go with the default name of adventurer, because I'm usually not willing to spend so much time thinking of a name I'm not going to be paying attention to in the future, and anyway, while it may be generic, "adventurer" does describe what the characters do pretty well, but I doubt you'd be asking for advice if you just wanted to do that.

Pathfinder.

My class name for a character that mixes melee and magik [that isn't Battlemage] because they stress Mercantile, Speech, Security and/or Stealth:

Insurgent

He's a mercenary-type who will be primarily working for the Blades and House Hlaalu if that helps.

Merc type I'll add; Myrmidon.

Others mentioned here for a recap;

Insurgent
Pathfinder
Adventurer
Ranger
Hunter


http://www.castles-of-britain.com/castle32.htm
http://www.svincent.com/MagicJar/Economics/MedievalOccupations.html#Military
http://freepages.genealogy.rootsweb.ancestry.com/~dav4is/Sources/Occupations.html Tabulated alphabetically. A very extensive list.

Some of the job titles are quite funny. Gong Farmer! Google that. :hehe:

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That character looks like a bounty hunter. There's plenty of synonyms: sellsword, investigator, mercenary, bloodhound, witchhunter (who says you can't make your own version of a pre-made?), tracker...