This is an idiomatic expression, not a phrasal verb. In this expression, the word mind means ‘consider’ or ‘remember’, a usage still current in Scots English, where one might routinely say, ‘Mind we're going for a beer later,’ or ‘Mind that game last Friday.’ (The BBC gives a couple of historical examples here). A familiar example is Auld Lang Syne:
Should auld acquaintance be forgot,
And never brought to mind?
‘Mind’, here, is ‘active, conscious memory’ (paralleling ‘forgot’).
The expression ‘Never mind’ still carries this sense. Its general sense is, ‘Don’t spend time reflecting on [whatever it was].’
A subtlety in this context is that ‘never’ means ‘don’t ever’. That might seem identical, and as a literal instruction it would of course be equivalent. In this case, however, the important idea is ‘don’t’, and it is intensified by ‘ever’.
The initial meaning of ‘mind’ here is something like ‘remember with focused attention’. ‘Never mind’ therefore ends up meaning something like ‘It’s not worth conscious effort.’
The force of the expression evolves like this, then:
Mind something (remember it and reflect on it)
Don’t mind it (ignore it: it isn’t that important)
Never mind it (do not ever waste your time giving it any attention)
This is all related to the sense of ‘mind’ in usages like ‘Mind the gap’, familiar to any regular user of the London Underground. This common announcement is an instruction to remember and attend to the gap between train and platform, when embarking or alighting.
In fact, this announcement reminds you: it puts something back into your conscious attention, just in case you were about to act without attending to it.
And if someone ever says to you something along the lines of, ‘But that was your turn, mind’ (a pretty common kind of construction in British English), once again ‘mind’ essentially means ‘specifically recall’.