Honestly? At about $1k you are likely to be a bit limited. I would nearly recommend a Macbook, but while editing doesn't typically use the video card like gaming does, I have seen more and more video suites require a dedicated video card lately. For audio editing though it is a decent choice. I personally have a Macbook Pro I use for my editing work. Originally I did my work on my workstation, but it died and the MBP has picked up the slack nicely.
However since in the older thread you bumped down there you mentioned Vegas, I am going to assume you are looking at PC notebooks as much as anything(And the gaming as well points to that).
I don't have any specific reccomendations off hand, though general things I personally do...
Go to a store and try it out first. In particular, as best as you can, listen to the thing. Audio editing in particular is great, till you have to fight the whine of an overly loud cooling system to hear your work and can miss details like that. Plus for me it is a personal enjoyment issue, to much noise from the machine just grates on me.
I personally would not look for a desktop replacement, not even sure you could find one at that price range. One they tend to be loud, and two the battery life sucks to the point of you might as well get a desktop at that point. Getting a discreet graphics card is not a bad idea, and if you are doing a good amount of compositing and effects then a decent one would be on the list. I don't personally think going for the top of the line, newest flashiest thing, will work to your benefit in this case myself.
Get plenty of Ram. The more the merrier in general. I do my editing on 2 Gigs on my Macbook Pro, and had a powerbook with 1.5 before that, upgraded from 512 which I did audio editing on for a long while. I would say it is a good idea to get at least 2 gigs of ram for video editing these days.
A decent processor, but no so powerful it causes the need for more cooling(See above). A Core2Duo seems to be the best choice these days, a shame as I would love to support AMD more myself.
A firewire port if you plan on capturing your own audio and/or video. I prefer a powered firewire port, but the only other laptops I have seen other than Macs with that I tihnk were made by Samsung, or maybe it was Fujitsu. Been a while since I looked into it. For some reason PC laptop manufacturers don't like to use the thing which is more than slightly annoying for someone like me as it means I would have to carry around more wall worts.
HD space. For this I would actually suggest looking at an external drive. I am personally trying out, and thus far have been fairly happy with, a 2.5" FW400 Bus powered enclosure with a Samsung 5400RPM 320GB drive. I have been able to run a large number of audio tracks syncd to a single video track with no problem on it, haven't tried it with a video editing project yet, but eventually that will come as I turn my attention to rebuilding my portfolio again.
If using an internal drive, get a decent transfer rate. A lot of times this means a 7200RPM drive, but that is not always true. I chose the above drive as I had a choice between a 200GB 7200RPM drive and a 320GB 5400RPM drive. Between the two I think most benchmarks placed my 5400RPM drive as being quicker on sustained transfers but they were pretty close no matter what. It is just because of the density of the data on the plates that helps with the transfer rate in my case.
And a decent screen with a possibility for an external monitor is not a bad idea either. For capturing audio I would strongly suggest getting an external interface card. Even a cheap one will work much better than the internal on pretty well any laptop. Depending on the quality level you are looking for, a USB Mic may suffice(AT2020 USB maybe?) but that depends on your specific needs.
Think I covered just about everything. Let me know if you have more questions on any of this. And of course I would bet you will get more comments from others.