It seems as though the button's switch contacts may be the problem. Looking at an image of the button taken from the ifixit Fujifilm FinePix S1 Camera Interface Buttons Replacement guide it appears to be a sealed unit wired into the button board assembly.
(click on image to enlarge for better viewing)
A search online has failed to find a replacement part. You may have to contact Fujifilm in your location to see if the part is available from them
If the button finally becomes unusable and you cannot source a replacement part, an option that you may wish to think about is using an app on your iPhone to take the pictures remotely. (Also for android phones)