Long ago, when something was yours and it started with a consonant (let’s say a boat), you called it ‘my boat’. When something started with a vowel and was yours (let’s say an elephant), you’d say… not ‘my elephant’, but ‘mine elephant’. This is fairly archaic language that only really exists in a few places anymore, but it’s very similar to the difference between ‘a’ and ‘an’.
The my/mine rule does have evidence scattered around history, though, particularly in naming patterns: You know the name ‘Nancy’? It comes from ‘Ann(e)’. Because parents would call their baby Anns ‘mine Ann’, which became ‘my Nan’, and then ‘Nan’ and ‘Nanny’ (the sense of ‘nanny’ meaning aunt or caretaker comes from the Greek word for ‘aunt’, ‘nanna’, but the sense of ‘nanny-goat’ comes from the nickname for ‘Ann’) and eventually ‘Nancy’.
Another name that got a similar treatment, though with much less permanence, was ‘Ambrose’. Ambrose became ‘Amb’, became ‘my Amb’, became ‘my Amby’, became ‘mine Namby’. Thus, the ‘Namby’ in ‘Namby-Pamby’ is referring specifically to someone named Ambrose.
But who? Well, a guy named Ambrose Phillip. He was a fairly well-known poet in the early 1700s, and also a politician. I can’t really tell you if he deserved the name ‘namby-pamby’, I mean, I didn’t know him myself, but what I do know is that a man named Henry Carey didn’t like him much, and instead of going up to his house and trying to beat him senseless like Americans do, Carey wrote a poem about it:
“All ye Poets of the Age!
All ye Witlings of the Stage!
Learn your Jingles to reform!
Crop your Numbers and Conform:
Let your little Verses flow
Gently, Sweetly, Row by Row:
Let the Verse the Subject fit;
Little Subject, Little Wit.
Namby-Pamby is your Guide;
Albion’s Joy, Hibernia’s Pride.”
The poem proceeded to become very popular, and, because it was critiquing Phillips’ penchant for florid prose, the nickname became associated with everything sentimental and insipid, frivolous, weak or silly.
The ‘pamby’ part of the equation, as far as I can tell, is fairly similar to rhyming things with words beginning with ‘schm-‘. So ‘namby-pamby’ is kind of like saying ‘Billy-Schmilly’, except with a lot more history.