Agreed on the refurb thing, that's definitely what I'd be doing if I was back in college. Actually, I'm thinking of doing it anyway because I think I need a mobile workstation that still supports W7 and they don't make those anymore.
Anyway, on to specifics:Screen size
Good/better/best: 15" 1920 x 1080
Size is somewhat negotiable, but 15 inch offers the best mix of portability and usability. Avoid 17 inch and 11/12 inch, but 14 inch is okay if portability is more important. For resolution, accept no substitutes. Unfortunately many websites and applications are maintained by lazy developers who assume everyone has a 1080p display so their stuff doesn't work well with less vertical real estate. Conversely, higher than 1080 burns a lot of battery life and doesn't provide much benefit.CPU
Good: Intel U-series i3/i5
Better: Intel HQ-series i5
Best: Intel HQ series i7
The U series is low power, dual core. HQ series is quad core, which is much better at handling real workloads. I don't feel right recommending a dually for more than light use so I'd strongly push you toward the HQ series, and mobile workstations in general. In theory an AMD A10 or A12 branded CPU would be a decent "good" alternative, but I see precious few of them in laptops that aren't fool-bait. They did create some Ryzen mobile silicon that's probably more competitive with Intel's midrange, and it'll be interesting to see if those make it into any decent lappies.RAM
Windows is a pig so going less than 8GB is going to hurt, but more than 16 isn't going to provide much benefit. They can always add more if they feel so inclined.Storage
Good: 500GB magnetic
Better: 500GB SSD
Best: 1TB SSD
A solid state drive is the single best performance improvement you can get, so it's very strongly recommended. The only reason I even included a magnetic here is that most of those aforementioned off-lease corporate machines won't come with one by default. The particular make and model is not terribly important, and likely won't be listed on the system specs anyway.
Good: Intel HD integrated
Better: Nvidia Geforce GTX 950M, 1050
Best (mech. engineer god): Nvidia Quadro x2000M or AMD FirePro Mx100M
Best (non-ME peasant): GTX 960M/970M, 1060/1070
Intel onboard doesn't suck like it used to, and can even handle a few games at reduced resolution. A basic discrete GPU will be a good step up though, and the gamers in the room will appreciate it. The reason for the split in the 'best' category here is that Quadro/FirePro are designed for use with engineering CAD applications like Solidworks, which are kind of a big deal for mechanical folks.
Price wise, looking over some HP/Dell/MSI/Sager options if you bought new you'd be looking at something like this:
Maybe you can negotiate a volume or education discount to get closer to your budget, or convince the budget owner to go up some since $300 is only going to get you the 'dog poop' tier.
Oh, one other thing to be aware of:
The high-end machine should be capable of running Creative Cloud, AutoCAD, Solidworks, etc. without complaint, and the midrange should be able to do so also but not as fast.
No laptop in the world is truly good at running AutoCAD/Solidworks, there's a reason that real engineers use big-boy multi socket tower workstations. I think our FEA/sim box cost about 20 grand. A mobile workstation, however enthusiastic I may be about them, is still going to be a lesser-evil choice. They'll certainly be a lot better at it than a $400 craptop though, which for an education setting may be sufficient.