What's your definition of budget?
FWIW I had to buy a laptop in a hurry for work (self employed and starting up broke) - I got a low end 14" Dell and while it was OK for the browser and office work I was doing the screen was terrible. Uneven colour and poor viewing angles and sub full HD made it feel like working through a letterbox.
I then replaced it with a more expensive 13" Dell with an IPS 1920x1080 screen as soon as I could afford to - my main aim was portability not photo editing as I have a desktop too.
Having a higher resolution on a smaller screen was a massive step forward.
The problem you're likely to have is that laptops are generally preconfigured so to get a half decent screen you'll need to pay for a higher spec processor etc.
The good news is that GIMP is quite relaxed in the specs it needs to run - though it will run better on faster machines.
I'd look for a used higher spec machine over a brand new lower spec machine and concentrate on the screen specs - at least Full HD or you'll be tearing your hair out.
If you don't need to lug it around then a larger screen will be better - 15" tend to be the sweet spot for price performance - but if you are going to need to carry it on trains, planes etc. then I'd go for a high res small screen model.