Though everyone seems to want to get rid of him (her?), I can't seem to make him. I dragged a nursing mom to the lagoon. And nothing. Does she have to be IN the lagoon.. or just near it? And what happens if you drag a mother nursing twins to the lagoon?

There are both puzzle and tech requirements. If you check the Virtual Villagers Guide, you can see what they are. As for the twins, you'll lose one of them.