A laptop that costs under $1000 will run EQ just fine if your willing to lower graphic details in some ways...low texture resolution, and perhaps cutting down a decent bit on spell effects and it should run beautifully...
...I play on a HP laptop that has a 1.9 GHZ AMD Turion 64 processor, 1 gig of ram, a 128 MB video card...and it runs very well, the only thing i have disabled is the sky graphics (you'd be amazed how much of a difference having those off can make) and also spell effects are cut down to nothing (never been much of a fan of them)...
...every other graphic detail, such as textures and character models, is set to the best...and i play with the maximum far clip plane (only time I've ever really lowered it is on a raid, when dropping it to 75% or even 50% can help alleviate some mild choppiness, though I've never had it get choppy to the point of affecting gameplay in any way. As for normal group gameplay, it runs virtually perfect.
one note...kinda obvious one anyways...don't plan on running any other major resource hogging programs in the background while playing EQ, it will drag down EQ horridly, of course...
I paid $1200 ($1000 after rebates) at Office Depot, though I'd recommend shopping around for the best deals...those rebates can save you some serious money, once you go through the pains of getting them sent off...
...one final tip would be that a laptop that has some expandability for another stick of RAM would be a plus, would allow you a quick way to get a solid increase in performance if you end up with the money lying around sometime in the future...
...anyways, good luck on the laptop!