Unless you want to specifically position the device at some intermediate location, I think a simple geared DC motor would work best, powered by a few NiMH or possibly Lithium rechargeable cell. Lithium's are more a bit more powerful per weight but are potentially more dangerous and cost more and are trickier to re-charge. Like NiCd cells, NiMH are simple to recharge. You can find ready-to-go NiMH battery packs and chargers for many toy-like devices, such as use in R/C cars.
A geared motor could be much smaller and use less instantaneous power, but would take a more time to travel than a direct-drive motor. The stage needs some sort of vertical bearing to limit the motion to vertical and not bind. A scissor lift could perform that role and be driven by a small motor. Also a single lever or two with a nut or threads on one end driven by a rotating threaded rod could do the job. This mechanism only needs to raise perhaps 6.25" but has maybe 13" of space to reside, so if you could build perhaps one single section of a scissor jack that should do the job. You might find all the parts you need with a robotics kit such as something from Makeblock or VEX robotics.
I imagine three 1/8" steel rods with what I can only describe as "soda straw" pieces about 1" length acting as the bearings for the stage. (Look into model rocket launch lugs to get the idea.) A longer length helps prevent binding, just as a wider axle for a bearing helps prevent side-wobble for more precise pivot-point mechanisms.
For starting/stopping, I'd suggest two lever-actuated limit switches, one at each end, to stop the motor when it reaches the travel limits. These can be wired up with a few relays to drive the stage up and/or down when the switches are pressed.
I'm wondering if you have given up checking for responses by now...