What criteria should I be considering when looking at a second hand sewing machine?
Basic requirements

For a basic sewing machine that does hems and maybe some simple garment sewing... I'd look for:


Basic stitches - running stitch, zig zag, blind hem stitch 
Adjustable tension - to accommodate lighter fabrics
Adjustable stitch length
Reverse (almost all modern machines have it, it's just handy)


Choices

If you want to sew buttons you also may want to know how the button holing works.  Technically you can make a button hole with what I just described, but many machines have slick ways to make the process easier or to make the button holes come out nicer.  If you think button holes will be a frequent task, make sure you like and understand how this feature works on this model.

Bobbin winding - every machine will have a way to wind the bobbin, but how easy/hard it is can be a real point of satisfaction for the user.  Make sure you like and understand how this process works.

Needle threading - most modern machines have a light near the needle to make this easier.  But also - how do you thread the machine - it's always more than sticking the thread in the eye of the needle.  If learning to thread the machine makes you want to cry, this is not the machine for you.  

Bobbin winding and needle threading are steps you'll do almost every time you sew (sometimes more than that!) so if you hate the steps or find them really hard, it's worth it to keep looking.

Similarly - every machine will have a presser foot and a way of lifting/dropping the needle - there's no right or wrong, but it's a comfort issue... like going on a test drive of a car.

Quality checks when buying


Be sure you can sew with the machine.  Unless you really like a good mystery, get a test drive of the machine.  Not only does it clarify the choices, but it lets you make sure the machine works.
Know and check that you've got all the feet and any other parts - check online what the model was sold with.  Be sure the parts are there: presser feet, button holing add-ons, the foot pedal controller and all wires, darning plate and any add-ons, spool pins or other add-ons for holding various types of thread and bobbins.  Personally, I wouldn't sweat a missing instruction book, but I'd verify I could get a copy online if I were a more novice seamstress.
Look at the shape of the machine - it should have a regular and even hum and the needle should go up and down easily without breaking, or slipping gears.  The needle should be able to catch thread in the bobbin easily, and not slam into the darning plate.  Changing settings should produce an obvious change in the outcome when sewing.