Removing Gaussian Noise from a Signal to Get Minimum Value
If you have no prior data on the signal of interest there is nothing to do actually.
The more prior you have the better you can do.
For instance, if the only information you have is the Bandwidth of your signal, the best you can do is apply an Band Pass / Low Pass Filter.
If you know a sparse representation of your signal it will be great, as you can remove all the other coefficients (Generalization of the LPF / BPF).
In case you know something about how smooth it is, you can apply Total Variation like smoothing.
Remark
When you write Gaussian Noise I assume you mean Additive White Gaussian Noise (AWGN).