hi there,
begonias propagate so readily from cuttings that it's probably only worth the trouble to grow them from seed if you're hybridizing or are trying to grow a species you can't obtain any other way.
that being said, begonia seeds are very, very fine (dust-like), and probably one of the easier species to grow from seed is B. subvillosa, which is covered in a brown fuzz but has rather plain white flowers.
for the color orange...'Orange Rubra' and 'Mandarin Orange' are two cane types that produce orange-y flowers.
other than those two, i can't think of too many orange begonias (except for the tuberous variety that don't do well as houseplants). are you sure you want to start your mom out on begonias? a goldfish plant is nice and easy and has orange flowers.

-ming

Thanks for the information, Ming. I'll check out the goldfish plant and the orange begonias you mentioned.