"Member FDIC" instead of "Member of FDIC"?

This "Member FDIC" is simply the shortest way allowable by law for a banking institution to indicate its affiliation with that organization. According to the relevant regulation:

328.3(1) Optional short title and symbol. The short title "Member of FDIC" or "Member FDIC," or a reproduction of the symbol of the Corporation (as described in § 328.1(b)), may be used by insured depository institutions at their option as the official advertising statement.

It is worth noting that legal regulatory language, especially in advertising, may be devised by organizations without reference to what may be customary for those businesses to use in communications. In other words, the whole issue stands at a remove from English proper. Note that the FDIC itself doesn't care whether the institution uses of in the construction.

In terms of grammatical justification, I believe it is a case of colon usage in lists, with a list of one (ie "Member: FDIC").

As far as why bother, a longer ad is a more expensive ad. When it comes to the legally required rhetoric especially, every fraction of a second shaved is worth the effort. It's given us those delightfully and craftfully edited legalese sentences that sound like how one might try to pronounce a paragraph typed without a space bar.