After Phoebe says to Holden, “You don't like anything that's happening...you don't like any schools. You don't like a million things,” she then asks him to “name one thing” he likes a lot.
Phoebe most likely challenges Holden to do this because
Phoebe realizes that Holden is bluffing, as she knows several things of which Holden is fond. Asking him to name one thing he likes is her way of proving to Holden that his complaints and negative attitude are baseless and unnecessary.
Though Phoebe is aware that Holden dislikes almost everything in his life, his bleak outlook upsets her. Challenging Holden to “name one thing” he likes is Phoebe's way of trying to prove herself wrong; she does not want to accept that Holden hates so many things.
Phoebe challenges Holden to “name one thing” he likes because she is trying to communicate to him that he is being unreasonable by rejecting everything in his life, just because he is missing and grieving for Allie.
Phoebe challenges Holden to “name one thing” because she is struggling to relate to her brother, but wants to; Holden's cynicism and Phoebe's childlike enthusiasm are clashing, and Phoebe is smart enough to attempt to find a way to understand Holden.
She thinks her actions will help Holden stop talking to Allie as if he were alive.