From the use of the noun burger as the second element in compounds denoting types of hamburger (e.g., cheeseburger), -burger has come to be used to form compounds designating persons or things characterised by the initial element.—Cf. also ‘Mc-’: prefix inspired by the McDonald’s restaurant chain.

NOTHINGBURGER

The compound nothingburger denotes a person or thing of no importance, value or substance, especially something which, contrary to expectations, turns out to be insignificant or unremarkable.

The first two occurrences of nothingburger that I have found are from the Hollywood gossip column written by Louella O. Parsons (née Louella Rose Oettinger – 1881-1972):

1-: Published in several U.S. newspapers on Monday 1^st June 1953—for example in The Philadelphia Inquirer (Philadelphia, Pennsylvania):

After months and months of wrangling, Sam Goldwyn ¹ finally has thrown in the sponge and given Farley Granger ² his release. But Sam retains an option to sign Farley again after an 18-month period.
Granger will live in Europe until the 18-month period is up and Sam either wants him or doesn’t want him. After all, if it hadn’t been for Sam, Farley might very well be a nothingburger. However, I have a hunch this whole arrangement is on a friendly basis.

¹ Samuel Goldwyn (born Schmuel Gelbfisz – 1879-1974) was a Polish-born U.S. film producer.
² Farley Granger (1925-2011) was a U.S. actor.

2-: Published in several U.S. newspapers on Thursday 5^th July 1956—for example in the San Francisco Examiner (San Francisco, California):

I can well understand why Shelley Winters ³ felt she had to explain to me that she had been misquoted and never had said the things against Hollywood that were attributed to her.
“I couldn’t deny them,” said Shelley, “because I was told if I did I would get in bad with the newspaper chain that said I never wanted to come back to Hollywood. I feel I owe this town too much to go against it.”
“You certainly do,” I told Miss Winters, who was Miss Nothingburger when Ronald Colman ⁴ gave her a chance in “A Double life.” ⁵

³ Shelley Winters (born Shirley Schrift – 1920-2006) was a U.S. actress.
⁴ Ronald Colman (1891-1958) was a British actor.
⁵ A Double Life (1947) is a film directed by George Cukor (1899-1983), starring Ronald Colman, the Swedish actress Signe Hasso (née Larsson – 1915-2002) and Shelley Winters.

MOUSEBURGER

The compound mouseburger denotes a young woman of unexceptional appearance and talents, regarded as timid, dowdy or mousy.

Its first known user was the U.S. author, publisher and businesswoman Helen Gurley Brown (née Helen Marie Gurley – 1922-2012), editor-in-chief of Cosmopolitan magazine from 1965 to 1997.

The earliest occurrences of mouseburger that I have found are from Helen Gurley Brown Only Wants To Help, by the U.S. journalist Nora Ephron (1941-2012), published in Esquire (New York City, N.Y.) of February 1970—this article was republished under the title “If You’re a Little Mouseburger, Come With Me. I Was a Mouseburger And I Will Help You.” in Wallflower at the Orgy (New York: Viking Press, 1970), a collection of articles by Nora Ephron:

“It drives my management wild to be compared with Playboy,” said Mrs. Brown. “We are not like Playboy. We are all the things we’ve been talking about—onward, upward, be it, do it, get out of your morass, meet some new men, don’t accept, don’t be a slob, be everything you’re capable of. If you’re a little mouseburger, come with me. I was a mouseburger and I will help you. You’re so much more wonderful than you think. Cosmopolitan is shot full of this stuff although outsiders don’t realize it. It is, in its way, an inspirational magazine.”
[…]
[…] She’s just worried that somewhere out there is a girl who hasn’t taken a bath during her period since puberty. She’s just worried that somewhere out there is a girl whose breasts aren’t being treated properly. She’s just worried that somewhere out there is a mouseburger who doesn’t realize she has the capability of becoming anything, anything at all, anything she wants to, of becoming Helen Gurley Brown, for God’s sake. And don’t you see? She is only trying to help.

The second-earliest occurrences of mouseburger that I have found are from Helen Gurley Brown: “Sex and . . .” author still peddling advice to single girls, by Tom Kaib, published in The Plain Dealer (Cleveland, Ohio) of Sunday 26^th April 1970:

“I work so hard because I’m still afraid if I stop too long I’ll fail. I started scared and always pushed.”
And that, really, is what Helen Gurley Brown and Cosmopolitan is all about.
“If you’re a little mouseburger, come with me. I was a mouseburger and I will help you.”