In which structure of the female reproductive system are eggs produced?

The correct answer would be ovaries.

Females usually have a pair of ovaries. Each ovary contains million of eggs since birth. However, only around 300,000 remain viable till puberty.

Out of these around 300-500 eggs are ovulated from ovary during each menstrual cycle.

Eggs developed in the form of follicles which are develops from primary to mature or Graffian follicle. It gets rupture and release ovum during ovulation day of menstrual cycle.