A white elephant is a possession that is useless or no longer wanted by its owner. It's also a high-maintenance item or property that brings little or no profit. According to old legends, white elephants were gifts kings would give to courtiers who were no longer in favor. The expense of maintaining the white elephant would financially ruin the courtier.
The baby Jig is likely carrying is carrying is a white elephant, as far as Jig's boyfriend is concerned. It is the kind of "gift" that he feels is not worth the cost. He wants Jig, therefore, to get an abortion, but he also wants her to go along with it. He wants her to at least pretend the abortion was a joint decision—and, of course, for it to happen, she has to agree. Jig, however, wants to keep a the baby.
The term "white elephant" also connects to the idea of "the elephant in the room," a big problem or issue that everyone in a room is aware of but nobody wants to talk about. Jig and her boyfriend keep talking around the abortion, not calling it by name. A further elephant in the room is their strained, angry relationship. Neither one, at this point, is willing to talk frankly about their problems. Their relationship is both a white elephant—no longer worth the cost of maintenance to either of them—and the elephant in the room they won't discuss.