Out of Pod Experience

I'm going to be going out of country soon for 6 months, and would like to purchase a laptop capable of running eve with 4 clients.

It doesn't need to be on high settings, lowest settings would be fine, will be doing mostly mining anyway.

I would really rather not spend much on this, say at most $300-500, because where I'm going it's very dusty and rough conditions, and the laptop is just going to get torn up either way. It just needs to last those 6 months.

From what I know, eve doesn't need a serious graphics card, but to run multiple clients would depend on the RAM, am I correct in this assumption?

I've looked and looked, and the laptops I've seen usually run around $500, but those come with integrated graphics, and I'm just not well educated about modern budget laptops and what they are capable of.

So, I'm here looking for suggestions.

Given the budget limits, it's questionable you could get a decent enough laptop with a discrete video card that can noticeably outperform the current integrated graphics.
You're probably have to resign yourself to integrated graphics.

The video part is more important for light grid loads, the CPU part matters more for larger-scale engagements.
Therefore, if you're mostly mining, you care much more about the video part than the CPU part.
In that case, an AMD fusion line CPU+GPU combo should be better for you than an intel processor with integrated graphics.

You're probably going to go with Win7 (because XP is a bit too dated already), so that means you'll have at least 1 GB (or maybe even at least 1.5 GB, since I think the video RAM is actually shared RAM in that case) occupied with the OS itself and other non-EVE things, and you really want the very least 500MB of RAM for each EVE instance (700MB or even 1GB preferable, but not critically required), so 4GB of RAM might just borderline do it, but 6GB would be highly preferable. There may be some swappage going on from time to time, but nothing too drastic if you minimize the memory footprint of each EVE instance. If you could get 8 GB of RAM you are home free though. Unlikely given the budget though.

So, I guess, borderline within your projected budget, a laptop based on a A6-3400M (or slightly better if it's not much more expensive) with 4 GB of RAM (or 6GB if you can find one that still fits the budget) will most likely be the best overall choice for your particular situation, and suffice for "playable" FPS with four miner-centric EVE instances with medium-low graphics settings.
It won't be very pretty, but, eh, with that budget...
For comparison's sake, the integrated video in an A6-3400M should be able to drive one EVE instance with high-ish graphic detail at HD (not fullHD) resolutions with close to 60 FPS in lightly crowded grids. Either way, the instance that gets focus should suck up more CPU+GPU power than the ones in the background, so I guess you could expect to get close to or even above 30 FPS at nearly minimal graphic detail at the minimum allowable resolution.

Thank you all for the very helpful information. Especially you, Akita. I ended up buying this.

Newegg

It's refurbished, and it just came in the mail today. Downloading the client onto it now, will see how it holds up shortly. It seems to surpass the minimums you set forth in your post so I think it will do fine. Also, it seems like a decent deal.

ebay: get an old one.

i say 'old', you can get decent laptops just a few years old for that sort of money easily able to crush 4 clients, heck even premium brands like alienware and gamer acers etc

Of course an old laptop may not be the most reliable, but as with all ebay stuff, just go with some one with a good reputable ebay record etc.

[edit] oh nvm.... glad you got it sorted though :)

Good catch with the refurbished deal, not sure how long it will last in the environmental conditions you described in the OP, but best of luck, hope it's a well-refurbished one that will not cause you issues. I assume you'll do some thorough and heavy-duty testing before leaving :)
I'm not sure adding an extra layer of air filters for when you're away would be the best of ideas (due to potential heating issues), but if you could at least manage to open it up and clean it out on a monthly basis (careful of static electricity, dry dusty environments are particularly wonky), it will probably last until after you come back and then some.
Just make sure you keep an eye on the temperatures, and consider a fresh cleaning whenever it starts to go up for no good reason compared to current values.

On the technical side, the Radeon HD 7660G inside the one you picked is roughly 45% faster than the HD 6520G inside an A6-3400M, so it should be able to yield 60+ FPS on highest graphic detail even in FullHD, not just the resolution the laptop is capable of.
You can probably run all four instances with medium-towards-high graphic detail at above 30 FPS, just make sure you upscale the options with the highest ratios between eye-pleasing effect to you vs GPU load. To test it out, do it with a single instance in a controlled environment (docked and default camera angle, plus another fairly fixed location with more complicated detail, like the outside of a station in a quiet system) and with vsynch off ("interval: immediate") to assess the actual performance, then turn vsynch back on when you're done ("interval: one").

Personally, I would recommend starting with something along the lines of disabled shadows, HDR and AA, low post-processing and LoD quality, medium shader, high texture detail - then fiddle from there.
Raise shader first if you want better image quality, then post-processing, then maybe add low shadows.
If you are experiencing a full memory and loads of swappage however, try to disable the resource cache first, only then lower texture detail to medium (low is just too gross looking IMO, and not that much of a big deal on memory savings vs medium).

This section assumes you're playing windowed mode (personally, I prefer it that way). If you play full-screen and are alt-tabbing between clients instead, it's not really relevant, it's as if you're already doing this.
The background clients get a much lower CPU/GPU priority even if they do get shown partially, so the one with the focus gets much better FPS compared to the rest.
To further minimize the CPU/GPU drain by the other clients when not urgently needed on screen, don't just leave them in the background, minimize them instead - they barely even register on the GPU in that case, and take only a bit of CPU time compared to having them in a window behind your active client.

As for the "CoolSense" thing, well, conceivably, it could be driving the fans differently based on temperature, so you get a low air flow (and therefore less dust in, and less wear on the fans, and less noise) when on low load, and high air flow when temperatures are spiking... but then again, I would have thought you could set such things firmware-wise in the BIOS, or you might not even need to set anything and the laptop does it automatically (hence the software might be redundant at best, or overriding better settings at worst).
Personally, I prefer all fans on full power all the time, but then again, I actually enjoy the fan noises, fan wear is not really something I'm concerned about, and I generally try to clean out the machine more or less regularly (and it is not housed in harsh environments to begin with), so, who knows... if it's not eating up much CPU/RAM, you might as well let it there.

The machine is unlikely to completely croak no matter what dusty environments you put it in, as long as you can bear them yourself to begin with and take a look at temperatures now and then.
A realistic bad but not worst case scenario could be solvable by disassembly, cleaning, and reassembly. Lasting catastrophic damage is unlikely if you keep an eye on the temperatures, and give it a de-dusting whenever it starts to get too hot.
From what I read, the A10-4600M has a safe operating temperature of max 100C, so if you can keep it under 85C most of the time, it will probably last quite a while... and with a 35W TDP for the CPU and GPU parts combined, it's not like there's all that much internal heat generation to begin with.
I'd be far more worried about potential water (or other liquids) damage ;)