Undergraduate finance is pretty much the same between Monash and Melbourne, I know alot of finance lecturers at both melb and monash - alot of them have taught at both melb and monash, the stuff taught is pretty much the same, and around the same quality between lecturers.
However, if you do want to continue further studies after your bachelors, then masters at uom is much better than monash - better reputation, better course in general and much higher standards.
If you've been provided a scholarship at monash, definitely take up the opportunity, there is marginal difference between the 2 at the undergraduate level and if you wish to continue academia after your undergraduate degree, just simply transfer to uom after you finish bachelors at monash, quite easy to transfer (along with the prospects of getting a scholarship) especially if you have top marks
What your mum said is partially true, employers would certainly "prefer" a melbourne graduate over a monash one since finance at melbourne is more reputable than monash, however this is only true at higher levels - if you graduate after bachelors, employers might even pick a deakin commerce masters graduate over you lol
So yeah basically in summary - definitely plan to do higher levels of study at uom, but for the 3-4 years of undergraduate study - pick the one you got the scholarship in.