After considering the question, I think I have an answer:
On the surface level, there doesn't seem to be much of a reason for Lisa to pick on Daisy at all. There are more unstable patients in the ward, and Daisy has never done anything to warrant antagonization in the first place. However, when considering how much Lisa craves freedom, the hatred becomes clear.
Daisy has the most privileges out of anyone in the ward: she's allowed to stay in her room any times she wants, receives gifts, and gets food from her father's deli- roasted chickens, while the others eat what the ward has for them. In contrast, Lisa is constantly monitored at every turn. Every escape attempt- no matter how long- has inevitably ended and left her back where she started. This boils up into a resentment for Daisy, and in turn, Lisa lashes out.
Lisa's hatred towards Daisy reaches its focal point when she and Susanna go to Daisy's new apartment in hopes of getting some money. Angered at the prospect of Daisy living comfortably...not quite free, but light years more so than her, hostility comes out.
It's almost eerie how excited Lisa is when dragging down Daisy, aiming to destroy her emotionally and mentally. This is also not mutually exclusive to Daisy either however: towards the end of the movie, Susanna is ensured to leave the facility...something Lisa can't handle. She ends up turning all of Susanna's friends against her, reading her diary.
This spitefulness against anyone she perceives as having more freedom than her is what leads to Lisa's hatred. An aspect that ironically, has also kept her in the ward for so long.