The earliest slang dictionaries to take note of "Miss Thing" (or "Miss Thang") are Geneva Smitherman, Black Talk: Words and Phrases from the Hood to the Amen Corner (1994) and Clarence Major, Juba to Jive: A Dictionary of African-American Slang. Here us the entry from Smitherman:
Miss Thang (1) An arrogant woman, one who acts high and mighty. (2) A derogatory term for a gay male who, through dress and behavior, over-exaggerates his femaleness.
And here is the entry from Major:
Miss Thing n. (1940s–1990s) derogatory term for male homosexual. (F[ield] Re[search].) S[outhern] C[ity] U[se], P[imp and] P[rostitute] U[se], P[rison] U[se], D[rug] C[ulture] U[se].
That "Miss Thing" has a fairly long backstory in Black American use seems clear from the fact that Count Basie put out a well-received jazz recording titled "Miss Thing" in 1939. However, in-person, in-print appearances of "Miss Thing" as a demimonde figure are elusive before the 1970s. One such appearance is in Nat Shapiro & Nat Hentoff, Hear Me Talkin' to Ya: The Story of Jazz as Told by the Men Who Made It (1955):
And here are some of the sporting women and nicknames of a few well-known Crescent City [that is, New Orleans] characters:
...
Also—Flamin' Mamie, Crying Emma, Bucktown Bessie, Dirty Dog, Steel Arm Johnny, Mary Meathouse, Gold Tooth Gussie, Big Butt Annie, Naked Mouf Mattie, Bird Leg Nora, Bang Zang, Boxcar Shorty, Sneaky Pete, Titanic, Coke Eye Laura, Yellow Gal, Black Sis, Boar Hog, Yard Dog, Bodidily, Roody Doody, Big Bull Cora, ... Tenderloin Thelma, Three Finger Annie, Charlie Bow Wow, Good Lord the Lifter, Peachanno, Cold Blooded Carrie, Miss Thing, Jack the Bear.
This list of names refers to characters from the Storyville District of New Orleans, presumably from approximately the 1900s to the 1930s or 1940s, as the discussion of sporting houses is in the form of reminiscences about the old days (as viewed from the middle 1950s).
In the context of gay culture, an entry for "Miss Thing" appears in an unidentified glossary in Anthropological Linguistics, volumes 13–14 (1971[?]:
MISS THING (n.): A name used by homosexual males for the purpose of convenience and anonymity (syn. MISS ELANEOUS, MISS GOUCH).
John Rechy, City of Night (1964), however, uses the name in connection with the guardian angel or gay fairy godmother of a character named Miss Destiny:
'Then Miss Thing said to me (Miss Thing is a fairy perched on my back like some people have a monkey or a conscience),' she explained, 'well, Miss Thing said to me, "Miss Destiny dear, dont be a fool, fix your lovely rair [red hair] and find you a new husband—make it permanent this time by really getting Married—and even if you have to stretch your unemployment, dont allow him to push or hustle" (which breaks up a marriage)—and Miss Thing said "Miss Destiny dear, have a real wedding this time."
By the middle 1980s, however, the term evidently had very pejorative connotations in the gay community. from Jennifer Blowdryer, Modern English: A Trendy Slang Dictionary:
MISS THING (n): A dragqueen insult. You can work the street all night, MISS THING, but you ain't never gonna afford all the plastic surgery that you need.
J.L. Lighter, Random House Historical Dictionary of American Slang (1997) has this entry:
Miss Thing n. Homosex. an effeminate homosexual man.—used as a joc[ular] or insult form of address, often to heterosexual men. [First cited occurrence:] 1972 B. Rodgers Queens' Vernacular 135: Miss Thing...affectionately effeminate.
Tom Dalzell & Terry Victor, The New Partridge Dictionary of Slang and Unconventional English (2006) emphasizes the association of the term with someone who is annoyingly arrogant:
Miss Thing noun used as a term of address for someone (female or homosexual male) with excessive self-esteem US [First cited occurrence:] You don't have to yell Miss Thing. —Hubert Selby Jr, Last Exit to Brooklyn, p. 58, 1957
Here is the excerpt from Last Exit to Brooklyn, at somewhat greater length:
Well for gods sake, you just going to sit there all night or are you going to tell us what happened. O really Miss Lee. Cant you see the poor girl is overwrought. You dont have to yell Miss Thing. Im simply dying to know what happened, thats all. Thats alright honey—O thank you Goldie—I understand. Just let me get myself together and I/ll tell you the whole story.
The gender of the person being called "Miss Thing" in this excerpt is unclear, and the sense of the sentence—whether "Miss Thing" is an allusion to the person's excessive self-regard or excessive effeminacy or something else is likewise unclear (to me). I also note that although Last Exit to Brooklyn is set in the late 1950s, it wasn't published until 1964, so the 1957 date given by Dalzell & Victor is somewhat misleading.
Jonathon Green, Cassell's Dictionary of Slang, second edition (2005) lays out the following evolution of the usage and meanings of the term:
Miss Thing n. (also Miss One, Miss Thang) 1 {1960s+} (orig[inally] US gay) a greeting to a fellow homosexual man [cross-reference omitted]. 2 {1970s+} (US gay) one's innate femininity. 3 {1980s+} (UK Black/campus) any unnamed woman. 4 {1990s+} (US Black) a woman who is seen as arrogant and unpleasant.
Because of the Count Basie and New Orleans jazzmen's allusions to "Miss Thing" from the first half of the twentieth century, I'm not at all sure that Green is correct that "Miss Thing" was originally arose in the gay community in the 1960s. On the other hand, the sense of the term in those early references is not at all cjear—and may refer to real people rather than to particular characteristics or attitudes.
In gay usage, the word might have some link to the Miss Thing at the center of J.M. Barrie's 1916 play A Kiss for Cinderella (1916), as obliquely suggested in Gage Lewis's answer. But I haven't found any direct link between the two. The closest thing to a unifying element is the fairy advisor "Miss Thing" in John Rechy's 1964 novel City of Night; but "Miss Thing" in Barrie's play is not the fairy godmother; she is the Cinderella character.
Green's assertion that "Miss Thing" in the sense of "a woman who is seen as arrogant and unpleasant" arose in the 1990s finds some support in published usage of "Miss Thing" in specific Black settings. Don Evans, "Sugar Mouth Sam Don't Dance No More," in Black World (April 1973), uses "Miss Thing" in a playful, not at all hostile way:
SAMMY. Ain't you still my baby?
VERDA. This one [glass of I.W. Harper whiskey] be for "my baby" ... An' the next one be for the road ... (Sings) "That long, long road..."
SAMMY. Alright, Miss Thing ... You make the toast ... It;s yours. (VERDA comes from behind the counter. They meet center.)
VERDA. (Raising her glass) ... Ol' times ... Let's make this one to "ol' times." (They both drink.)
A brief item in Jet magazine (September 20, 1979) suggests that the negative sense of the term wasn't fully developed at that date either:
How innocent-imaged Melba Moore has changed her song content. On her upcoming album, she wrote a sex-dripping tune titled My Love Is Hot and Tasty, whose lyrics are very suggestive. Her pre-album release, however, is the less risque Miss Thing, which addresses itself to the leather jacket crowd.
I'm not sure what demographic "the leather jacket crowd" refers to.
Ten years later, another Black singer had come to be known as "Miss Thang." From Donald Ades, "Mills Takes Us 'Home' from 'The Wiz'," in the [Whittier, California] El Paisano (December 7, 1989):
Just when Patti La Belle thought it was safe to release a new album along comes "Miss Thang," as Patti lovingly knows her, with a new album that will undoubtedly spoil her Christmas.
Well, Patti, girl, I hope you're worried because it seems that folks are going to be bringing Stephanie Mills (Miss Thang) "Home" for the holidays this year.
News reports say that LaBelle and Mills were close friends, so this "Miss Thang" nickname was not an instance of LaBelle throwing shade.
Ntozake Shange, Spell Number Seven (1985), however, has one of her characters use the term in a seemingly insulting way:
BETTINA. well miss thing with those big ass hips you got/ I dont know why you think you can do the ballet anyway
The Company breaks; they're expecting a fight.
And going back to Lorraine Hansberry, A Raisin in the Sun (1959), we have this striking instance:
BENEATHA. Mama, you going to take that [an old house plant] to the new house?
MAMA. Un-huh—
BENEATHA. That raggedy-looking old thing?
MAMA. (Stopping and looking at her) It expresses ME!
RUTH (with delight, to BENEATHA) So there, Miss Thing!
Conclusions
The evidence regarding the origin of "Miss Thing" is too scattered and incomplete to give a definite sense of exactly where it originated and how it evolved. Hubert Selby has the term being used in a milieu involving gay drug users of different races in the late 1950s. Lorraine Hansberry has it being used negatively in a striving working-class Black milieu in 1959. Other early instances going back to New Orleans in the early decades of the twentieth century are sketchier and therefore harder to interpret.
What seems most probable to me is that the expression developed in different tracks—one positive or neutral and one negative—in both Black culture and gay culture. Although it may have been a thoroughly derogatory term in both cultures by 1994, it also seems to have had some moments of positive use over the decades, even as late as 1989.