As in "I wake up, do my hair, have breakfast". Is there a verb for this? I'm guessing you can't just use 髪をする.
As pointed out in the comments by @chocolate, 髪をセットする is a good way to say "to do one's hair". It can involve blow-drying, applying hair wax etc.
髪型をセットする means "to fix one's hairstyle", which sounds reasonable if you always wear the same (elaborate) hairstyle.
If you are among friends you can also say 寝癖を直す as @永劫回帰 suggests, meaning to get rid of the unnatural bouncy waves coming from sleeping. (As people wash their hair in the evening and usually not in the morning, this is a thing in Japan, especially for people with shorter hair.)
We say simply "do my hair" "髪を整える." We don't say "髪をする."
"寝癖" when it refers to hair, means the hair deshelved while you were sleeping, and "寝癖を直す" is used specifically for dressing your hair that became disheveled over night.
"髪をセットする" is more often used when you have your hair style set in a beauty parlor, like "今日美容院で髪をセットして貰った."