My 16 year old has frequent headaches and her aunt (my sisters) suffered a brain aneurysm at age 40, should I let her have an MRI now to check?

Possibly. Brain aneurysms are acquired over time with the average age of rupture, in both genders, of age 52. It is very unusual in teenagers and people in their twenties. The likelihood is low, and there may be actually other reasons that could cause the headaches that would warrant imaging of the brain. Please see the brain aneurysm center for women for more information at www.Chicagoaneurysm.Com.
Probably not. Headache from aneurysm rupture is sudden, severe (worst headache of your life), not typically frequent headaches. Aneurysms that run in families (familial) affect parents/siblings/children, less so with aunts/uncles/cousins. Have your teenager evaluated by a headache specialist.
Yes. There are genetic aneurysms. Mri does not radiate your child. I had one done on my own son for the exact same reason. I would want nothing less for your child.