There seem to be very few restrictions.
12 USC § 241 - Creation; membership; compensation and expenses:
In selecting the members of the Board, not more than one of whom shall be selected from any one Federal Reserve district, the President shall have due regard to a fair representation of the financial, agricultural, industrial, and commercial interests, and geographical divisions of the country. The members of the Board shall devote their entire time to the business of the Board and shall each receive basic compensation at the rate of $15,000 per annum, payable monthly, together with actual necessary traveling expenses.
So, you need them to NOT share a Federal Reserve district, you need them to be a diverse representation of various economic interests and geographic areas, and they can not hold a competing job (shall devote their entire time to the business of the Board).
The president can clearly pick pretty much any bumpkin as long as Congress will approve them. Case in point: Sarah Bloom Raskin (the last one in the Wiki list of governors - I'm sure I can easily find more if I bother checking the rest of the list).
Nominated by Pres Obama in 2010.
Her education is J.D. degree from Harvard Law School and a B.A. in economics from Amherst College (not exactly a stellar economics academic achievement - no MA, never mind PhD in Economics). Her only listed relevant experience is 3 years as Maryland state banking regulator. Commentary after the appointment was that she's expected to basically just serve as a yes-person to Ben Bernanke, it'd be interesting to check how that worked out.
- Contrast her with another Obama appointee, Janet Louise Yellen - an American economist and professor, who is the Vice Chair of the Board of Governors of the Federal Reserve System. Previously, she was President and Chief Executive Officer of the Federal Reserve Bank of San Francisco, Chair of the White House Council of Economic Advisers under President Bill Clinton, and Professor Emerita at the University of California, Berkeley's Haas School of Business. Decidedly NOT a random bumpkin.