Any idea the surface area of a solar panel required to charge a hearing aid battery?
Probably very small. Without knowing the battery type and recharging demands, impossible to know.
Assuming it's a 3 volt lithium ion rechargeable button cell ...
The panel will need to put out 4.2 - 4.5 volts at 100 milliamps up to 300 milliamps, but not more than that ... It will also need a circut to stop charging if the battery heats up too much or gets saturated and fully charged.
For a 1.5 volt cell, you'll want to charge at a maximum of 2.5 volts, 100 - 300 milliamps.
The higher the milliamp rating, the faster the battery will charge, but the more heat that will be generated. Some batteries can handle 500+ milliamp charge rates (fast charging), but some can explode with only 200 milliamps if they are small and delicate.
If you have a store nearby like Staples, Batteries Plus, Target, or something similar, you can ask for more detailed information concerning your battery and home charging needs.
Apple will soon have an iwatch. Where you wear it is unknown. I thought about a hearing aid style, or behind the ear maybe. Curious as to what it would take to make it solar powered.