To change a bike from flat bars to drops is not cheap and is not worth the expense. So a decision to be made as to a flat bar bike or drops. Dudelsack has already made the point about riding on the flats on drop handlebars.
There are womens specific bikes (Normally designated WSD) that are manufactured for the female form but they do not fit or suit every girl. Only way to check is to test ride and it could be that a normal mens bike fits better.
Most of us agree that choosing the make of bike is not a problem but finding the right Local Bike Shop (LBS) is. The good LBS will listen to your needs-Offer you a few choices and then ensure that you get the right bike for you and that it will fit correctly. You will not get that from a department store or mail order OR from a bike shop that just sells bikes to get them out the door.
So first suggestion is to find that LBS. The brand of bike you buy will be dependant on what the LBS carries and there are plenty of good bikes out there So get out in your locality and window shop for the LBS that is going to treat you right and get the right bike for Mum.
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How long was I in the army? Five foot seven.
Spike Milligan