A common technique relies on filter banks and robust statistics. The idea is to isolate some frequency subbands where the transformed signal is sparse, and the rest is (filtered) noise. From here, you can use a median estimator.
In the context of orthogonal wavelets (a form of dyadic filter banks), if the signal is sufficiently sampled, the highest frequency subband often satisfies the above requirements. Or one can choose a suitable union of subbands where the signal is considered sparse.
Then, if $c_i$ denote the subband coefficients, an estimator of the Gaussian noise variance $\sigma$ is:
$$ \hat{\sigma} = \frac{\textrm{median}|c_i|}{0.6745}\,.$$
The following picture displays the average noise estimation, with standard deviation, from several realizations of a Gaussian noise added to a line of an image.
This is described in Penalized threshold for wavelet 1-D or 2-D de-noising. More details are given in S. Mallat, A wavelet tour of signal processing, section 11.3 Thresholding Sparse Representations,
Noise Variance Estimation. The denominator factor is discussed in Rousseeuw and Croux, Alternatives to median absolute deviation, 1993.
A very basic avatar consists in differentiating the data with the basic $[1,-1]$ finite difference, and using the above estimator, further divided by $\sqrt{2}$ to account for its energy normalization.
