It is helpful to be clear about what you are trying to measure. The frequency response (aka transfer function) of the string will be given by
where $T(f)$ is the frequency response, $I(f)$ is the input signal, and $O(f)$ is the output signal and all three are a function of frequency. In order to have a meaningful frequency response function, it is important that $I(f)$ and $O(f)$ have meaningful units, which is achieved through proper calibration of your drive and sensor.
One difficulty in your situation is that you don't have a good measurement of $I(f)$, so you have to make some assumptions about it's responsivity to your electrical drive. Since audio frequencies are much lower than typical piezo resonances, you can probably assume that the frequency response of the piezo element has a flat amplitude, but the phase may start to roll off at higher audio frequencies. If your goal is simply to measure the amplitude of the $T(f)$ and ignore the phase, then you should be safe simply driving the piezo with white noise and taking the $O(f)$ to be the amplitude of the frequency response function.
In a perfect world where both your sensor and transducer have no noise and no saturation, then a quick white noise measurement is enough to fully measure the frequency response of any system. In the real world, you have to be careful of the following (I'm sure I'm missing some):
- Saturation of the peizo
- Saturation of the microphone
- Saturation of the electronics driving the piezo
- Saturation of the electronics reading the microphone
- Producing a sound which is below the background noise
- Producing a sound which is below the noise level of the microphone.
- Nonlinearities in the instrument causing response at frequencies other than the driving frequency.
Some example reasons that you might want to use a signal other than white noise:
- Measuring at discrete frequencies (swept sine) allows you to make higher SNR measurements at the specific frequencies which you are interested in because you can integrate for longer.
- If the response is very low at some of the frequencies you are interested in, then your sensor may not pick up the sound. In this case, you may want to drive harder in certain frequency bands (pink noise).
- On the other hand, if the response is very strong at certain frequencies, it may drive your sensor into the nonlinear regime which will destroy the measurement. In this case you may want to drive less in certain frequency bands (pink noise).