Suitable motor for pic WAKING project
Your main problem is to have a stock of dry pillows.  

A motor for this is overkill. You don't want to actively rotate the bucket, leave that to gravity. Better to balance it against some rest, so that removing the rest will topple it. You can use a solenoid for this, or maybe a servo (but then we're using a motor again).  



To solve the humidity problem: don't fill the bucket with water, but when you balance it, don't tie it, so that it comes down itself. No dry pillow stock needed. :-)

edit
Since you want a slow release of the water you'll need a motor to control the rotation. Use a DC motor with a reduction gearbox. You'll find those on any robotics website.  



Another solution, which is mechanically simpler is to use a electric water valve; the bucket can then remain in place. You can find such a valve in an old dishwasher or washing machine. The water valve will probably release too much water, but you can reduce this to drop level by closing the outlet, just leaving a small opening, so that there will only be a few drops released.