It’s the lack of RAM and the graphics that’s killing the performance on your system. I don’t have an recommendation for magical free or cheap computers that will run it, you just need to find some money or a sponsor and get some machines that exceed the minimum specs, and stick some good graphics cards in there. Have you asked the school? Sometimes they have funds specifically for computers that must be spent by a certain date or forfeited. I see this all the time around here.
There are some decent deals on refurbished systems out there, but honestly you may be better off buying new or even building custom systems.
My school district buys direct through HP and there’s a small discount, but it really isn’t a great price. You could build a better system for cheaper than we buy through HP.
I’d recommend a dual-core processor over 2GHz, 4 Gigs of RAM for general work, and 6 or 8 gigs for a full robot model, and a workstation-class video card from the nVidia Quadro or ATI FireGL or FirePro line. The cheapest video card to run it well is probably $100 alone. The RAM and video card is really going to make the difference. People have “softmodded” Geforce cards into Quadro cards, but I have no experience with that. Talk to Adam Heard, as he has done it. Or, you could go with one of the Intel SandyBridge Core iX series chips with the integrated Intel HD 3000 graphics which is pretty decent, but that isn’t cheap either.
I’d recommend a system with DDR3 memory. And make sure you’re running Windows 7 x64.
I came across this in the past, but I have no real knowledge of it http://computersforlearning.gov/
Also, I recall BestBuy has grants I think for $1000 that you may be able to apply for.
I quickly threw some hardware into a shopping cart on NewEgg. For a complete system worth of hardware, the cheapest Core i3 with Intel HD 3000 graphics, and 4 gigs of RAM, the total comes to $315, and I’d guess SolidWorks would run just fine on this system. The price includes case, power supply, motherboard, RAM, hard drive, and CD/DVD burner/reader drive. However, this cost does not include Windows 7 x64 which would run you another $100-$130 per machine roughly, depending on if you went with home or pro.
So, figure $450 for a new system to run SolidWorks. This is the cheapest system I’d be comfortable running it on. By the time you add in the cost of Windows, you don’t save a whole lot by building your own system vs. purchasing something turn-key. However, you have to be careful with turn-key systems, as none in your price range will have Quadro or FirePro graphics and few will have Intel HD 3000 (not HD 2000) graphics.
For laptops, I bought myself one of these on sale for $450 a few months back: http://www.frys.com/product/6677364?site=sr:SEARCH:MAIN_RSLT_PG and it’s a better machine than the desktop system I describe above, for the same price (and it has a screen, which the price I quoted for a desktop did not include). You could probably find an even better deal in a 15.6 inch laptop; I wanted a 14 inch specifically. I spent $35 for 8 gigs of RAM, and it runs a full robot model in Inventor 2012 just fine. I have not tried SolidWorks on it. Deals are out there if you keep your eyes open.