There's no relation between menstrual cycle and navel piercings, however there are important issues regarding age. Even if the local laws are more lax, it is not appropriate to have a navel piercing if you are under 16 years of age, as the body is still developing. A piercing that looks aesthetic if done when you are too young, can end up in the wrong spot by the time you are fully grown.
Also, navel piercings take 6-9 months to heal! So they take a lot of patience and maturity to deal with and care for during that time. For example, you can't go swimming at the beach while you're healing (unless you cover your piercing with a waterproof bandage).
Contrary to the above there is a correlation between the cycles and the healing of a navel piercing. More often than not about 2~ 4 days before the cycle arrives the piercing can become annoyed and some what cranky, discharging a bit more than normal.
This is often mistaken as an infection and treated with various products all of which just annoy the piercing even more so. Simple irrigation of the piercing while moving the jewellery under warm running water in the shower eases this discomfort and once the cycle starts the piercing reverts to a more normal behaviour.
The actual healing time for a navel piercing to be considered seasoned is a full year. These statements are based on over 18 years of hands on experience in the body piercing field.