I have only one door bell on this summer home. The button shows power as it is lit. When pushed, the only thing heard is a clicking noise in the chime box which is located approximately 20 feet away. I checked the wires of the button and cleaned the contact points, then re-connected the wires. Could my problem be the chime box? Or possibly something else...

It is quite possible that the problem lies in the chime box.

Inside there is a solenoid that is energized when the button is pressed. This shoots a plunger into the chime and makes the "ding". When the button is released, the plunger retracts quickly and hits a second chime, making the "dong". If the plunger does not freely move, this could prevent the bell from working. It could also be a broken spring in there. It should be obvious once you're in there.

Here is a fantastic graphic showing what I mean:
http://www.knockdoorbells.com/advice_main.htm

Good luck,
WallyH

Some times you need to remove the plunger, lightly wipe with WD40 and 0000 steel wool and use an ear swab to clean the tube the plunger rides in. If all else fails, the lower end chimes are only $20 or less. It could be the transformer but I would bet plunger as WallyHelps says.