At this time I don't think the Fed Funds Rate matters much re banks cost of funds. They're mostly paying essentially zero on checking accounts. (Once upon a time, people all had savings accounts and checking accounts, and the savings accounts earned interest and the checking accounts didn't. That line blurred and was essentially erased, so now checking account interest rates are what matters. After that, rates fell, and people's expectations of receiving interest just evaporated.)

This is an amazing chart. I tried plotting the 3-month T-Bill rate on the same chart, but the numbers are so different that when plotted on the same scale, the checking account rate curve can't even be seen. I've included the text below the chart, to make it clear what this represents. Yes, it is really an average of what is being paid at all banks.
Sure there are a few banks paying significant rates, ala 2% for some savings accounts now. Those banks are using the high interest rate as advertising. In other words, instead of paying an ad agency to run a lot of ads on TV and whatnot, they just post a high interest rate, and let a zillion bloggers and bank industry hangers-on (bankrate.com) do the advertising for them.
Rates have been low so long that the public at large has forgotten all about interest on bank accounts. These few little banks advertising high rate accounts are not yet causing significant money to flow out of accounts at traditional banks. Until they do, banks have no reason to significantly raise rates.
I think that it will take SEVERAL YEARS before banks are paying reasonable rates on checking accounts. They're all gonna play chicken and raise rates very slowly, watching the rate of customer complaints and/or withdrawals. This should allow banks to be very profitable for the next few years. That's why I've bought bank stocks.