Seth Masket doesn't follow my logic regarding women in a supersized House. He notes that state legislatures are not "teeming with women," which I suppose is true, but about a quarter of all state legislators are women, while just over 15% of the current House is female.
Looking it up....the numbers aren't quite as impressive as I expected, but they do suggest that new Members are disproportionately female (compared to all Members, not to the overall population!). So, ten of the fifty-four new Members at the start of the 110th House were women; five of the thirteen Members seated during the 110th House were women, so that's 22%. In the current 111th House, it looks like exactly one-fifth of the new Members so far (including replacements) are women (I'm using Wikipedia and the invaluable CAWP for the numbers). On the other hand, however, recent groups of new Members have been mostly Democrats, and Democrats are a lot more likely to elect women to office these days than Republicans are.
All in all, I think it's very likely that if we suddenly tripled the size of the House, somewhere between a fifth and a quarter of the new Members would be women, which would be at least in my opinion a significant increase.