If your budget is under $500, I'd recommend prioritizing a laptop with a solid CPU, an SSD, and at least 16GB of RAM. That combination works well for many PC titles and classic console emulation.
As someone who runs a PlayStation-focused website I've found that older games and emulators often perform better on balanced hardware than on a machine with only a stronger GPU. Has anyone tested newer budget laptops with PS2 emulation? I'd be interested to hear your experience.