Form FBAR says that if one holds more than $10000 in a foreign bank account at any time during the year, one has to file the form.
If I am sending $12000 to "my" foreign bank account and then give the money to my mom right away, do I have to file Form FBAR?
I was advised before that I don't have to file Form FBAR if I am not keeping the money in the foreign account and giving it to my mom right away.
Since I don't know exactly when the fund will hit my foreign bank account, the money may stay in my foreign account for a day or two before it is sent to my mom's account.
In this case, do I file Form FBAR?

Yes.
United States Treasury Regulation 31 CFR 103 require that every United States person file this form TD F 90-22.1 if at any time during 2006, you have an interest in or a signature or other authority over a financial account in a foreign country, such as a bank account, securities account, or other financial account and the total amount at any time in the year was $10,000 or more. Read: Mandatory Reporting of Foreign Bank and Financial Accounts (FBAR)