The only time I've heard this is in the sense of "choose me." When the teacher asks the class something like, "Who wants to raise the flag today?" or "Who knows the answer to the question on the board?" particularly eager children, in addition to waving one hand in the air, might irrepressibly exclaim, "Pick me!"
Adults have begun to use the phrase ironically as something of a catchphrase, said in a sarcastic tone of voice.
"I hear management wants somebody to straighten out that mess in the filing system."
"Oooh, pick me! God, that would take at least three months."