Easy. DC Rebirth is about re-establishing character history? Well then let Power Girl remember that the Superman of Earth-2 (either pre-Flashpoint or New 52 version is fine) was basically a father to her, not just a cousin, as well as that Lois Lane being basically a mother to her. That alone gives her some separation from Supergirl for new readers.
Have her meet up with the Kent family. Have it be a bit awkward between them, like maybe Rebirth Superman remembers meeting the Earth-2 version during Infinite Crisis (they've confirmed that did happen in Rebirth, right? I mean he does still remember Superboy Prime, as shown in an issue of Nightwing). Obviously he can't fill the same role as his counterpart and both he and PG know that, but he still wants to help her and she still misses having a family (this can help lead into the revival of the JSA, which DC has been hinting at).
Let her bond with Jon Kent way better than with Clark and Lois, because she's never met a version of him. Let her be like a big sister to him. Someone he can turn to for advice, like the kind he can't get from his parents or Damian.
Boom. Power Girl has been re-established back into the DC universe. She has connections with Superman and his family but they are shaky because of coming from two different worlds (not an uncommon thing in the Superman line of books) and let the writer go off from there.
But, and this is most important, DO NOT make Power Girl just an older Supergirl or a sidekick. Make her more like Nightwing. Someone who used to live under their mentor's wing but is now a hero of their own making. Someone who is proud of who their mentor/father figure was but not just a copy of them.
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