The problem with the pronounciation is all down to alphabetical letters used in medieval scripts that looked like the number '3'. When typeface printing came along, these weren't included in the printer's sets of letters, and it was usually represented in type as a 'z', it being the nearest shape.
But...the way the '3' was originally written, fully rounded, or flat top and rounded base, for example, could all vary the pronounciation of the letter. It could represent a 'yee' sound, or a hard 'g', or a soft 'z'
Compounding this is the name itself, which is found over Scotland from 'Mingus' (Min3uss, where '3' = hard 'g' ) through to 'Mennies' (from Men3iss, where '3' = 'yee' ).
(Most Scottish place-names containing a 'z' retain the 'yee' sound, as in 'Dalrulzion' = 'Dal-rul-yion).
The result of all this is that you'll just have to ask any person called 'Menzies' just how they want their name to be pronounced.