Misti- I presume she can do homework in the kitchen, in either of the 2 downstairs living rooms, or in her shared bedroom either in the evening before 6 year old goes to bed when he is downstairs, or quietly with a desk and lamp when he sleeps.
We have a desk in bedroom as well as living area and dining table to work at. When both children are asleep I still work from desk in bedroom if Dh is working louder ie conference calls in living area, and he does the same if I have calls. But the general using laptop or paper and pens we do in the same room as children when they sleep.
I understand she may not like it. But that's the space they have. She's 16, and surely can appreciate that sometimes life isn't perfect, but she ist see also that her life is 100% better than so many if the only issue is sharing a bedroom.
I would suggest she asks for some adjustments in the way everyone works in the house together. For example if 6 year old goes to bed at 8pm, can she ask her parents that they make sure he isn't in the bedroom 6-8pm. That way she gets 2 hrs every evening to work alone or do whatever.
Same with other things. In a shared bedroom 6 year olds toys need to primarily be in living space so they aren't all over the place in bedroom, and he will stay in living areas more playing with them. Just books, teddies and odd small toy in bedroom. Also saves on space.
There are many things that can be done to improve living which are based on space, but on curtesy of living under one roof