This appears in some dictionaries with the sense prim or affected, over-refined or mincing.
Aficionados of Lewis Carroll will know a different meaning, which appears in the poem called Jabberwocky in his Through the Looking-Glass: “All mimsy were the borogoves”. Later in the book, Humpty-Dumpty explains its meaning as being a blend (he calls it a portmanteau word) of flimsy and miserable, so meaning “unhappy”. Carroll either invented it afresh or borrowed an existing English dialect word and gave it a new meaning.
In the sense of affected or over-refined, mimsy has long been known in the British Isles, especially in Scots and northern dialects; an example is in A Rock in the Baltic, by Robert Barr (1906): “In one corner of the room stood a sewing-machine, and on the long table were piles of mimsy stuff out of which feminine creations are constructed.” It’s known in other spellings, such as mimsey and mimzy; mimp is closely related; an elaborated version is miminy-piminy or niminy-piminy.
All forms seem to be built on mim. This little word may come from an imitation of pursing up the mouth in prudishness (a related form is mim-mouthed, affectedly prim and proper in speech, which appears in Virginibus Puerisque, by Robert Louis Stevenson, published in 1881: “Mim-mouthed friends and relations hold up their hands in quite a little elegiacal synod about his path: and what cares he for all this?”)
Mimsy is far from dead. I found it in the issue of The Medical Post for 6 January 2004 (which was published in Toronto, but the writer was remembering his childhood in Scotland): “Certainly if I had been drafted into the Armed Forces I would have been streets ahead of these mimsy Boy Scouts with their cowboy hats and their two-fingered apology for a salute.” It also appeared in an article by Griff Rhys Jones in the Independent on 24 October 2003: “This is food writing. Not mimsy pseudo-porn, but genuinely funny gastro-investigation driven by a slavering appetite.”