Perhaps one of the biggest examples in theater is William Shakespeare's Falstaff, a buffoonish companion to Prince Hal in the Henry IV plays. The plays were intended to celebrate Henry IV, while Falstaff is written as a poor influence who must be shunned once the prince matures. Despite Falstaff's negative characterization, he proved a fan favorite. The audience's sympathy for the character is evident in Henry V, where his death is described in heroic terms. Finally Shakespeare decided to fully cash in on Falstaff's popularity by ripping him out of his previous continuity and plopping him in modern day Elizabethan times to star in his very own comedy, The Merry Wives of Windsor. An apocryphal story holds that Queen Elizabeth asked Shakespeare to write a play about Falstaff in love because he was her favorite character.
Let's just say that Shakespeare's true intention behind Hal's rejection of Falstaff, and which of the two is meant to be the hero, has been the subject of fierce debate ever since. (The hero is definitely not Henry IV, though; despite being the title role his part is very small.)
There have been reworkings of Henry IV which embody both parts one and two, that are named "Falstaff" the entire play is built then with Falstaff for the most part playing the jolly Pinball Protagonist
Shakespeare produced a few more darkhorses in his various plays:
Mercutio is the darkhorse of Romeo and Juliet. As the witty comic relief, he gets the lion's share of good lines before his death marks the play's turn into tragedy.
Many people adore Benvolio, if only because he has a cool name. In many screen adaptations, he has a far larger role than in the original — he even got to be part of the Beta Couple in Romeo X Juliet!
The witches in Macbeth certainly qualify, to the extent that some scholars believe several of their scenes (particularly those involving Hecate) were added by somebody else after the play was originally published and they had been established as popular characters.
Arthur Miller's play Death of a Salesman is considered one of the greatest American plays — not at all for the reasons Miller intended, but he knew why. Looking back, he wished he'd focused more on the character Biff, the protagonist Willy Loman's son.
A real-life example that catapulted the actor's fame: Miss Marmelstien, a piece of Christmas Cake who laments her lack of a beau from the little-known musical I Can Get It For You Wholesale. The character sounds like, and was intended to be, a supporting character - until they cast Barbra Streisand in that role, her first Broadway role, and she stole every scene she was in. Supposedly her last line of the show won her a standing ovation.
In the hands of a skilled actress, the Shark girl who sides with the men in the number "America" can be this and a One-Scene Wonder, though it's hardly applicable for every performance.
Mystere has the Red Bird (aka Firebird), who participates in the Korean plank/trampoline/fast track act but is primarily a dancing character who weaves in and out of the action. By 2006, this character was popular and recognizable enough that the show got a new logo that featured it, as you can see at the show's trope page.
La Nouba has a bird character of its own, the kooky dancer known as the Green Bird; she was upgraded to logo status around the same time the Red Bird was.
Also, Archie deserves special mention, even though he's a bit more of a main character. Although this really depends on the actor.
A disproportionate amount of Starlight Express fan art and fan fiction centers on Electra's components, who are tertiary when compared to most of the cast.
The most popular character in Clifford Goldsmith's play What a Life was a very minor one, a dorky teenager named Henry Aldrich. The character of Henry got such rave reviews that NBC Radio decided to adapt What a Life into a radio series with Henry as the protagonist.
Enjolras and Eponine are two of the most popular characters in Les MisÚrables even though they aren't introduced until midway through act one and are both dead midway through act two.
The ACT Theatre production of A Christmas Carol has this with the "Turkey Boy", who (of course) delivers the prize turkey to the Cratchits at the end, but is introduced much earlier than in the novel. Also, Mrs. Dilber the laundress (who also acts as an opening narrator in this version) and the "plump sister"(identified as Ms. Clackett here).
In First Date: A New Musical, recently premiered by the aforementioned ACT in collaboration with the 5th Avenue Theatre, the Waiter gets his own spotlight number in the form of "I'd Order Love".