Girl Friday

Christine Everheart: After all these years, Tony still has you picking up the dry cleaning.
Pepper Potts: I do anything and everything Mr. Stark requires. Including, occasionally, taking out the trash. [still smiling] Will that be all?

—Iron Man

The faithful female servant. She's not an Action Girl, but neither is she a Damsel in Distress; her main job is to help The Hero (her boss) out at critical junctures. Sometimes she provides critical information, other times it's just a helping hand or heart, sometimes with a side of Unresolved Sexual Tension. Don't presume the Girl Friday can't fight however — she can still kick plenty of ass, sometimes turning into a full-fledged Battle Butler.

The term is the Distaff Counterpart of "Man Friday," an expression that comes from Robinson Crusoe's "man Friday" (man as in servant, Friday being his name); it was popularized for female office workers by His Girl Friday, a Screwball Comedy about a reporter's relationship with her editor/ex-husband. Interestingly, neither of these Trope Namers exactly fits this trope.

Compare with The Jeeves, Beleaguered Assistant, and Hypercompetent Sidekick. A step up from the Sexy Secretary.

Examples of Girl Friday include:

Anime and Manga

Kosuna from Desert Punk mostly fulfills this role to the titular Sunabozu, at least, until she becomes the main character herself.
Shirakawa in Elfen Lied. Unlike her boss Kurama, she's as heartless and ambitious as anyone else of the researchers.
- In the manga. In the anime, she seems to have somewhat of a conscience, restraint, and a heart, with the head researcher Isobe getting the role of a heartless, cold bastard in her stead.
Riho from Nightwalker was this for both Shido and Yayoi. Once she gets turned into a vampire, though, she turns into something more for Shido.
Mimi from Mnemosyne is this for Rin.
Gender flipped with Kosei Hida, Meifon Li's Bishonen assistant from Angel Links.
Eirote to Seness in Scrapped Princess.

Comic Books

Pepper Potts in Iron Man, who is becoming more and more badass. She even has her own armor now.
Both Shellie and Gail have performed this role for Dwight McCarthy in Sin City. Gail does get her hands dirty from time to time, though.
Mercy Graves for Lex Luthor — at least, as seen in Batman: No Man's Land. Her duties range from bodyguarding to business to serving dinner.

Film

The Trope Namer is, of course, the 1940 Screwball Comedy His Girl Friday, although the character of Hildy Johnson (played by Rosalind Russell) doesn't quite fit the trope as we define it.
Kaylee from Big Fat Liar is essential for all of Jason's scams. She fakes voices, helps out the legwork, strategizes etc.
As seen in the image, the version of Pepper Potts portrayed by Gwyneth Paltrow in the Iron Man movies very much applies.
Shelley Barnes, the lesbian assistant of Jack Gramm, in 88 Minutes.
Though the character of Lucy wasn't George's secretary in Two Weeks Notice, he certainly treated her like one so much that she was pretty much a personal nanny to him.
Miss Moneypenny. "Oh, James! Do be careful!"
- Note that Moneypenny is technically one of these to M, rather than to Bond himself.
- In the novels, Mary Goodnight was Bond's Girl Friday, and unlike the film of The Man with the Golden Gun, she was actually competent. She was the secretary for the 00 agents and they had a wager to see which one of them would seduce her.

Literature

Margo Lane and The Shadow.
In Robert A. Heinlein's Stranger in a Strange Land, Jubal Harshaw has three girls Friday.
- Not an example though with the title character of Friday, though she has great loyalty and affection for "Boss", her usually-unnamed (some know his real name, Hartley M. "Kettle Belly" Baldwin) father-figure employer who uses her as an Action Girl courier in the Divided States of America.
Leisl is this to Sasha to some extent in the Ravenloft novel Vampire of the Mists, although it is strongly implied that they get the a Relationship Upgrade by the end.
Effie Perine, Sam Spade's assistant in The Maltese Falcon.
Velda, Mike Hammer's Sexy Secretary. Digs up information, carries a gun which she uses on a couple of occasions, Unresolved Sexual Tension.
In Atlas Shrugged, Hank Rearden has Gwen Ives, who along with being his secretary, helps him run the administrative end of his steel business.

Live-Action TV

Cordelia Chase begins Angel as one of these before becoming an Action Girl.
Natalie and Sharona from Monk.
Chloe from Smallville.
The West Wing had a whole fleet of these girls: Donna, Margaret, Bonnie, Carol, Cathy, Ginger and Mrs Landingham (who also had a Girl Friday of her own, Nancy).
Gail Koner of Profit, who takes a level in badass in "Chinese Box" to become Jim Profit's Battle Butler.
Betty Suarez (Ugly Betty) is this for Daniel Meade.
Special Agent Diana Barrigan of White Collar, while a full agent herself, often plays this to Peter.
As Veronica Mars has the High Concept of "high-school Noir with all the genders flipped", its Girl Friday role is filled by Wallace Fennell.
On Parks and Recreation, April has this type of relationship with Ron as his assistant. But since Ron's job consists of sitting at his desk doing nothing, April's explicitly stated duty is to keep him from ever doing any work. Since she's equally adept at avoiding work, their partnership works out perfectly.
On Fringe, Astrid has elements of this in her usual role, but in the Storybook Episode "Brown Betty", Walter casts her as the full-fledged version to Olivia's Hardboiled Detective.
Guinevere from Merlin is often this to Merlin, especially in the first few episodes.
On Queer as Folk, Brian's assistant Cynthia is clearly the person he depends most on in his work, though he later starts depending more on Ted as well even though he is technically only his accountant (perhaps because he knows Ted privately as well as professionally).
Most of the Doctor's female companions in Doctor Who.

Video Games

Cortana from Halo is an interesting variation.
Another (quite literal) Girl Friday is Nigel's Fairy Companion Friday from Landstalker.
Maya Fey/Ema Skye and Trucy Wright for the Ace Attorneys Phoenix Wright and Apollo Justice, respectively.
- Miles Edgeworth unwillingly gets one in his own game, in the form of Lady Thief Kay Faraday.
- To some extent, the detectives are these to their prosecutors. So Ema Skye is also one to Klavier seven years onward.
Yeoman Kelly Chambers serves in this role in Mass Effect 2.

Western Animation

Batman Beyond: This is what Terry would like Max to be, but she's a little too Action Girl to be content with it.
Janine is all over this trope in most incarnations of the Ghostbusters franchise, but particularly in The Real Ghostbusters and Extreme Ghostbusters.