How does the Brita LED work?
If an average faucet flows at 2 gpm (according to the US EPA), and the filter does not significantly reduce that flow, then 100 gallons is about 50 minutes. 

An inexpensive  CR2032 is rated at 230mAh, which is more than enough. Chances are it's something even cheaper like a stack of a few LR44 button cells. That can also be used to retain information. 

The other options (electrochemical action, paddle wheel generator, miniature nuclear reactor) would be too expensive or would cause contamination.