Hi,
Yes, there is a way to make her stay, what's more if you unlock her during the party, she can appear during other days, not only in Friday parties (thing is, the interactions with her are kinda limited to only bj rn).
To unlock her (convince her to stay overnight) you'll have to balance your drunkenness and impression as well as have some pretty good base stats, being tipsy gives bonus points to impression and base stats, And you actually need both to convince her.
If you manage to convince her, then you can somewhat try to summon her for the night using the "rain stick" that can be brought in the shop.
Hope this helps,
Cheers