In the comics, Hope became the villainous Red Queen who had essentially the same powers as her mother. Other than that, she's a relatively blank slate, so they could take her character in all sorts of different ways.
Something else to take into account is that there are incarnations of the Hank Pym character that make him abusive towards his wife, Janet. At the moment, Janet has not been confirmed to appear in the movie at all. Is Hope's relationship strained with her father to the point where she would go by Hope Van Dyne rather than Hope Pym? Is her mother perhaps dead to begin with?
The plot is supposed to revolve around Pym acting as a mentor to Scott Lang and instructing him to pull off a heist with the Ant-Man technology to circumvent the villainous goals of Darren Cross, who will be a member of the company taking it in a more evil direction. Lilly is without a doubt the love interest for Lang, and there's a chance that as much of a love story element might be between them, there may be an even bigger story between her and her father.
It's no secret that conflict drives films, so what if Hope has a controlling stake in her father's company and is aligned with Cross until Pym redeems himself with the help of Scott Lang, who is in need of guidance that Pym gives him in return?
I still maintain that even as solid of a story as that might be, that it is a mistake in the long run rather than going with a generic origin tale of Hank Pym and Janet Van Dyne as the young protagonists. You can even keep the casting decisions the same and just rename the characters and tweak the script. I will trust that Marvel's team knows what they're doing, judging by their overall track record, but that doesn't mean that they haven't made mistakes before. This idea of Hope Van Dyne could be much worse, but it could also be so, so much better.