Someone who everyone complains too, or someone who complains or whines constantly. Background info: my dog will suck my thumb whenever she needs comfort. When she does this, I say "Awww, do you need to wean?" (It's like she's trying to get some milk, because she was improperly weaned when she was a pup. Don't worry - she's fine now). So then whenever someone whined and needed comfort, I'd think, "do you need to weeeeeeeean?" Hence, a weanstick is a person who attracts or is prone to weaning.
So -n- so is a total weanstick, always whining and seeking attention, needing to be comforted.