It could either be the (visible) button that needs to be replaced or the power button switch (small circuit board). Both can be ordered, e.g. on ebay (button and switch).
However, before buying anything, I would check which part is actually needed.
To check the switch: Remove the switch/circuit board of the power button and lay it on the table (the contact or "eye" on top). Take a pencil and use the side of the eraser, slightly pushing on the contact several times. You need to hear a "click" or at least "feel" the "click" to be sure that the switch is still ok. If it is still ok, I would inspect the power button. Your issue seems to point to a worn out piece, either the switch or the button.
Use this guide for replacing the power button, and this guide to get access to the power switch (see last step).