Faucets is probably the worse way you can "earn" a little bit of bitcoin. Anything else using your computer's computing power (you mentioned keeping your browser open) is also a bad idea because as CPU mining has been unprofitable for over 18 months now.

If you have any kind of coding or similar computer skills I would recommend going to the marketplace subforum (services child board) to try to find some kind of work that you can do for bitcoin.