What to look for in a processor for a DSP project
The FFT is a popular benchmark for DSP chips, so it should be easy to find performance data relevant to your application. For example, if you're sampling audio at 44.1 ksps and doing nonoverlapping 512-point FFTs, you need to be able to do about 100 FFTs/second. Yes, it's true that a fast ordinary (non-DSP) microcontroller could probably achieve this level of performance.

The amount of data going into and coming out of the FFT is the same, so if you're using 16-bit data, you need to be able to sustain a rate of 88.2 kB/s when writing to the EEPROM.

Doing both of these things at the same time continuously can get a bit tricky, but the details are left as an exercise for the student. It never hurts to have some extra CPU bandwidth.