Elaborating on the first definition, I believe this came up in an attempt to find a socially acceptable and fast way for guys to talk about a female that they would like to have a real relationship, none of that one night silliness.
Opposite to wanting to "tap that ass," "hit that shit," etc.
One guy to another: "See that cute girl in the yellow dress playing frisbee? I'd like to hamrel with her."
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