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!
Give the user a bonus even for such a low amount he's gone to such great lengths, now getting serious there's a simple solution get professional service, you could use ipqualityscore check all IP address and block people having same address, also yiu could use cron to do it, just set it up and it'll take care. It's a one time thing and should solve your problem.