Test for dead pixels (if you can return it with even one dead pixel). You'll want to stress out your monitor as much as you can to see if you got any, since few appear once the monitor is stressed. To do so, set it to maximum contrast, and let it warm up for a while. You should warm it up with both white and black screens (depending on the type of monitor, black will stress it as well). What I mean by "stress" is that you want to give it some heavy use, so that the LCD has to go through their full range of motion. So bonus points if you can manage to make the monitor flash alternating black-white for a couple of hours or something.
Then to see if you have any dead pixels, set your monitor to black, red, green, and blue. For the black screen, see if there are any bright dots (of any color). For the other screens, see if there are any black dots (which is gonna be hard, since you have to get close to tell). Since (most) LCD monitors are made up of red, green, and blue units, this is pretty much a complete test for dead pixels.
Otherwise, if you got a DVI-capable monitor, look for the DVI cable. Look for the warranty card or something along those lines as well, so you know what terms you got. Maybe adjust the monitor's colors to your liking. Watch some movies on it to gauge the color quality and response time. Make sure you're satisfied with the viewing angle (LCD monitors look different when viewed at different angles). Beyond that, don't know what else off the top of my head.