Not so long ago, we simplify registration process at our site leaving only the username field...
As a consequence, we are faced with a problem of creating multiple accounts by the same user and abusing faucet (faucet at our site - 0.007 mBTC and only request every 3 hours when balance is zero).
Guys, what solutions do you recommend?

Thank you!
Lower the faucet amount and at the same time put max claims per day and should not based on hourly limitations because it is really prone on abuse.Set max claims per IP so that those abusers would really have a hard time on abusing faucet this is not new anymore and you are not only whos experiencing such problem but most sites do experience this such abuse thats why strict limitations would really be needed.
I agree with you to lower down the faucet so that it won't attract those people who hunt from faucets. I have been in different kind of dice sites and see people who try their luck in faucets so its better to require the players to deposit minimum amount every week or monthly. And also keep track to people who logs in with the same IP.
If you'd lower the amount people could withdraw from the faucet, you would also inconvenience the people who are using the faucet legitimately.
It's just a bad solution to an easy problem to solve.
For example, Crypto-games has one of the highest faucet payouts, they just require you to level up to get to the highest payouts.
Requiring people to log in would even be better than just lowering the payouts.
There will be abuse, but not impossible to sort out manually if needed.