Personal Mook

Jack Slater: Sir, are you a henchman?

Benedict: No, I only go as far as lackey.

—Last Action Hero

Sometimes, villains have henchpeople dedicated to particular tasks.

Such as:

They will have more personality than your conventional mook and may even be The Dragon.

Examples of Personal Mook include:

Comic Books

Sin City has Manute who started out as Ava Lord's bodyguard, driver, and The Dragon.
- Also, even though Blue Eyes is a mook assassin herself, she often uses a lumbering mook named Gordo to do a lot of heavy lifting.

Big Bad Adrian Ripburger in Full Throttle has two henchmen, Bolus and Nestor, who act as his bodyguard and his driver, respectively.
Big Bad Goldfinger had Pussy Galore for his personal Pilot and also as the instructor for his own personal airforce.
- Stromberg from The Spy Who Loved Me had his personal helicopter pilot Naomi.

Scarface the World Is Yours has lackeys that range from highly intelligent NPCS to ones the player can control. Calling for a boat or a car will gain a driver that will follow and kill for you. There are also the Assassin, Driver (note the capital) and Enforcer. All have differing tasks...if you choose. All can assist Tony (the player) in multiple ways, such as slaughtering bothersome gangs or making cash. Oddly, the last three can kill regular citizens, unlike Tony. Many of the NPC mooks, like most citizens of Miami, have unique dialogue with Tony, this giving character development.
Evil Genius gave the player character a colorful personal bodyguard once they reached certain levels of Notoriety.
World of Warcraft Cataclysm gives us the Lumbering Oaf, which serves Helix Gearbreaker, acting as a guardian and mount for the evil little goblin. The Oaf must be defeated as phase one of the Gearbreaker boss fight, and even has his own special move where he throws a fully voice-acted tantrum (which is odd for unnamed enemies) and slams a player into the wall.
The Godfather 2 allows you to recruit these for your own Badass Crew. They have special skills that will come in handy above and beyond their superior stats to the standard Corleone mook/redshirt.

We have Hammerhead's unnamed Chaffeur(ess?) in The Spectacular Spider-Man.
- "Drivey"
- Hammerhead himself starts out as a Dragon-flavor of this to Tombstone, seeming to exist solely to relay the boss's orders to the mob so he could maintain his front as a Villain with Good Publicity. After Tombstone starts questioning his competence, however, Hammerhead becomes The Starscream and, ultimately, the Green Goblin's Unwitting Pawn.
Lex Luthor of Superman: The Animated Series had his valet Mercy.
Owen Burnett is David Xanatos's general assistant on Gargoyles. This is because Owen is actually the fairy Puck, who first approached Xanatos in this guise and offered him a wish. Xanatos, recognizing the competence and usefulness of the Owen persona, wished for a lifetime of faithful service from him.
- He was impressed enough that he never tried to pull any of his usual tricks on Xanatos.

World Without End's prospective antagonists have their own, Godwyn has a lick spittle, Philemon, who has no real uses other than another person to make hear Godwyn talk. Ralph has the more useful Alan Fernhill. Ralph seems to be the older version of Alan.