This was actually originally a female name. It is an Anglo-Norman feminine form of Elias. It went out of use, but survived as a matronymic. It became popular in 16th-century England to use the mother's maiden name as a first name, and that is how it became a masculine name.
According to the data from the Social Security, 18 % of Elliotts born now in the US are recorded as women. The usage should probably be changed to unisex. [noted -ed]
Over-given in France for several years, as Eliot, Elliot, Eliott, Elliott, etc... Both for little boys and... pets (true). Looks weird for a girl, from my point of view.
I really like this name for a boy or a girl. But it's very popular in France at the moment, as a boys' name. I know at least 6 boys called Elliott, all under 8 years old.