Where does the word “minge” come from?

Your question was actually the subject of a paper: Minge --- A Loanword Study by James Clackson. Clackson asserts that minge is, like pal, nark, and lollipop, a loanword from Romani Gypsy. While the OED dates the first use of minge in text as 1903, it had been used since the 19th century in (very) informal conversation --- it belonged almost solely to spoken English, and even then, appeared to have been commonly used only by speakers of certain dialects.

However, to complicate matters, the origin of minge in the Romani language itself is somewhat uncertain. Clackson suggests that it came into the Romani language as a loanword from Armenian.

The origin of minge is rather interesting --- I suggest you read the paper. It should adequately answer your question.