Genetics. at least in my case. And I've doomed my 2 dd's the same as dh is rather hirsute himself (we joked about taking shares in immac should we have kids when we first got together). Dd1 (6) and dd2(2) are already both showing signs of inheriting the dark body hair gene from me. Thankfully they are not aware of it yet but it's only a matter of time. Both boys in the other hand have fair body hair.
Anyway, to try and ease some worries... By all means get checked out by a gp. Especially if no one else in the family is particularly hairy. If they are, then helping your niece accept this is who she is is essential.
Whilst it wasn't necessarily the best thing to do, from an early age I wore trousers and long jumpers (even through summer). Finding some pretty, longer length clothes in the short term might help your niece feel less self conscious of herself. Lightweight trousers for school, thin cardi to cover her arms. PE is going to be unfortunately either accepting it or asking if joggers can be worn.
I completely understand how your niece could feel. It is only since meeting my husband and having children that I've been able to relax in my own body more. And even then, I have to shave every other day. Waxing might buy me another day or two where I can pluck random breakthrough hairs out. But then I start looking like a yeti again. Thick, dark hair on legs, arms. Even a 'tache (plucked).
However, what I will insist on is if other children are picking on your niece because of this, it needs stopping now! Your brother has to go into the school and let them know how this is affecting his dd. let her know that it is nothing to be embarrassed about and that she can tell someone if she is getting teased. Whilst it is an extreme, no one would tolerate teasing if it was because of the colour of her skin. Whilst it's a different issue, the feelings left by the teasing and bullying are the same.