Determining Power of White Noise Component of an Incomplete Time Signal
I assume you have information about the signal itself (Not the noise).

What you can do is use LS to estimate the signal (Easily compensate for the missing samples in the Model Matrix).
Once you have it estimated you can do 2 things (Which are the same):


Recreate the samples using the model. Subtract the measurements. You'd be left with IID Samples of the noise -> Use the empirical STD / Variance estimator.
If it is Linear Least Squares use its internal estimation of the noise Variance.


Good Luck!