BPM Generator with display?
This could easily be done with a microcontroller. 240 BPM is 4Hz. By using a microcontroller with timer peripherals it would be easy to get accuracy in the +/-0.01% range. 

Suggest a microcontroller such as PIC or AVR or MSP430 with a small display, buttons. An LCD display could be used to save power, and there are micros with LCD controllers on board. Cost and circuit complexity would be very low.  

You can prototype this with something like an Arduino. 

Whilst you could use a 555 and attach a frequency counter to it (and adjust the knob as it drifts off frequency), the frequency counter would best be implemented with a micro and thus it's easier to simply synthesize the frequency you want correctly in the first place. 

You can find open-source LCD module display frequency counter designs based on the PIC16F628, for example, but they're probably not directly usable for such low frequencies. To get 1 BPM resolution with a simple frequency counter requires a 1 minute gate time, so period counting and math would be a better approach- actually more difficult than generating a set frequency.