The origin of the phrase comes from Daniel Defoe's novel Robinson Crusoe:
In a little time I began to speak to him, and teach him to speak to me; and, first, I made him know his name should be Friday, which was the day I saved his life.
The character Friday in Crusoe is a man--a savage that Robinson encounters on an isolated island. However, Friday becomes a faithful assistant and becomes invaluable to Robinson.
Nowadays, a Girl (or Guy) Friday is an all-around office helper. However, the phrase can apply to anyone who is skillfully versatile and dependent in many tasks.