The story
From the phrase pick me + girl, the implication being that such women, like someone pleading "pick me!", are desperate to be chosen over their putative competition.
Etymology from Wiktionary, CC BY-SA 4.0.
pick-me girl
Today it means: A woman who asserts that she is unlike (and sometimes better than) most other women, in order to gain attention, approval, or validation from men. Full definition →