Using white noise to test filter freq. response?
Yes you can also identify a system (more spcifically an LTI system) by inputing white noise to it and measuring the corresponding output. Yet this will be a statistical approach compared to the sine-wave measurement which is deterministic.

First, you have to be able to generate a WSS (wide-sense stationary) white noise of power $\sigma_x^2$ as the input $x[n]$ for testing.

Then you have to measure the output signal $y[n]$ and estimate it's power spectral density (PSD), assuming it's also WSS (which is indeed). So you will use some statistical approaches to obtain those quantities.

Then the relation between the input PSD of the white noise of power $\sigma_x^2$ and output PSD of the (colored) noise through the LTI is:

$$ S_{yy}(e^{j\omega}) = |H(e^{j\omega})|^2 S_{xx}(e^{j\omega}) $$

where $S_{xx}(e^{j\omega})$, and $S_{yy}(e^{j\omega})$ are the discrete-time input and output PSDs respectively. Since $S_{xx}(e^{j\omega}) = \sigma_x^2$ the output PSD will be:
$$ S_{yy}(e^{j\omega}) = |H(e^{j\omega})|^2 \sigma_x^2 $$

However, what is worth noting here is that,  this method yields not the frequency response $H(e^{j\omega})$, but its magnitude square. So the phase information is lost.

A better approach, which yields the frquency response $H(e^{j\omega})$ directly is to use cross-correlation or cross-power spectral densities instead.
The relation between input noise $x[n]$, output noise $y[n]$ can be expressed as

$$ S_{xy}(\omega) = H(\omega) S_{xx}(\omega) $$

Which shows a relation between input PSD and cross-PSD, and reveals $H(\omega)$ directly when input PSD is known as in the case with white-noise input PSD of $S_{xx}(e^{j\omega}) = \sigma_x^2$:

$$ S_{xy}(\omega) = \sigma_x^2 H(\omega). $$

Here the cross power spectral density, $S_{xy}(\omega)$ between the input $x[n]$ and the output $y[n]$ can be obtained through the DTFT of the cross-correlation between $x[n]$ and $y[n]$ from

$$\phi_{xy}[m] = \mathcal{E}\{x[n]y[n+m]^*\}$$ 

Which is indeed equivalent to the impulse response of the LTI system to be identified :

$$\phi_{xy}[m] = \sigma_x^2 h[m]$$