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!
I am sure that there is a way to prevent someone to do that thing especially if you are admin of the site. You will not have any problem to detect people who are trying to creating multiple accounts and abusing faucet. You can make some limitations for every user especially for people who are coming from the same IP, you can reduce the rewards of the faucet, or if someone caught abusing the faucet, then you can ban his IP. I am sure that you will find a good way to make sure your site secure from the abuse.
I think you are still making people punished for what the abusers are doing. Yes people who do not invest at all and do not deposit to it usually are not the type of people you want however back in the day when I first started gambling couple years ago I loved the faucets because there was a real chance that I grind that to withdrawal amount without depositing any single cent to it, that hooked me to the game, of course I have never managed to do that but whenever I lost I got closer to depositing and eventually deposited one day and ever since than I deposit and play with real money. Faucets was the one that got me hooked, if it was play money back in that day maybe I wouldn't be a gambler today.