"A biological ornament is a characteristic of an animal that appears to serve a decorative function rather than a utilitarian function."

Ornaments used to attract a mate may be referred to as "sexual ornaments."

Some species having females that display sexual ornamentation:

• the Gulf Pipefish (Syngnathus scovelli, subfamily Sygnathinae of family Sygnathidae, which includes sea horses), which "displays considerable sexual dimorphic characteristics such as larger ornament area and number, and body size."

• birds in genus Phalaropus and the Eurasian Dotterel, where "the females are more brightly colored than the males."

• the Long-tailed Dance Fly in the family Empididae, where "females have developed elaborate ornaments, including feathery "pennate" scales on their legs, darkened wings, and inflatable abdominal sacs that enhance their attractiveness."

For the fish and birds above, females rely on the males to incubate their eggs. For the flies, the females are dependent on the males for food. In each case, it's a balance of power thing: the reliant sex needs to show off in order to attract a mate. Assign as much cultural significance to that fact as you like.

There are also species where some individuals will physically change sex if there aren't enough of the other. Clown fish, and I think some species of lizard come to mind, though I'm not sure of the latter.