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!
You can't. There is absolutely no way to protect that aside from IP restriction. Which would actually help out because its harder to spoof your IP compared to just logging out and logging with another account. Even tho you can make it harder for them to abuse it, if you are giving out FREE money to people in the end there will always be people who will try to abuse that right.
However if you make the faucet small enough and the withdraw amount big enough just like primedice than it would be harder for them to withdraw and makes it useless. So, two things you can do to at least slow it down and make it tiny ; make it IP restricted and make it 100 times lower than the withdraw amount.