What is the female equivalent of the morning erection that some men experience?
Women have erections too! These erections are called clitoral erection. Clitoral erections are usually accompanied with vaginal lubrification.

Just like men (see here), the absence of norepinephrine during the REM phase of the sleep causes erections. In women, this phenomenon is called Nocturnal clitoral tumescence while it is called Nocturnal penile tumescence in men. In women, not only the clitoris get engorged with blood but the vagina too.