No minimum is probably the most important to me. Because this generally means no fees even if my savings dip below a certain amount. There should also be no fees up to a certain point -- for example, savings accounts should let me withdraw from it once in a while without incurring any fees.
High interest is nice, but of less importance to me. This is mostly because I already invest my money in the markets. Having a 1% risk-free return would be great, but chasing returns is more what I use my money for in an investment account, not as much in a savings account. So high-interest is a 'nice-to-have' for me but definitely not a requirement. It should, however, most definitely have a higher interest rate than a checking account.
For me personally, Ally.com works as a savings account since I mostly just use it to store money, and since I almost never withdraw from it, I don't have to worry about fees.