Hello, I would simply choose the second option, because, first of all, users doesn't like to login everytime they would like to claim, it'll be a pain.
Also, if you don't keep your users entertained they may leave your website, to keep your website entertaining add some games where they can earn some satoshis, a chatbox where they can chat with others and (of course) not a lot of annoying ads a/o pop ups.
And if your website is working well, you can add an Ad Slots renting system where users and advertisers would be able to rent the ad space for an amount of time a/o impressions.
And finally, add some new and unique things, this will help your faucet grow!

Regards,
Houssem.