Takes 6 bits of counter and some and gates (4 inputs worth), but if you count to 111100 on those 6 bits, read them out into and gates, and pipe that into the counter reset pin, you should have your solution. Will count 0 to 59 and will reset to 0 on 60.
That being said, there's a limit to how much accuracy you're going to see using normal parts. The 60Hz from the mains is a great time reference, not as good as GPS, but quite good... but when you're adjusting your resistor to change the output frequency, you're relying on your own calibration point and the stability of the parts. For better stability and repeatability, you want low tempco resistors and a high stability, low leakage cap like a polypropylene (as mentioned). Then it's just up to how closely you can do the initial calibration of the RC time constant.