Is it possible to add some kind of noise to a non stationary (having unit root) signals to make them stationary?
One processing approach is to use the Two-Sample Variance (also called the Allan Variance), which takes the differencing over intervals of increasing duration from which it can be directly observed over what interval the original signal (unprocessed) can be assumed to be stationary, and for other durations this processing can convert non-stationary signals to be stationary for further statistical evaluation. This is used widely in the Clock world due to the non-stationary characteristics of oscillator phase noise, but is applicable to other signals as well that have similar non-stationary noise processes (1/f noise, drift, etc).
More information on the Allan Variance (or it’s square root the Allan Deviation, ADEV) is here at Wikipedia:
https://en.wikipedia.org/wiki/Allan_variance
Differencing is a high pass comb filtering response, and works in the case of converting non-stationary clock frequency noise since the non-stationary aspects of the noise are more pronounced as you get closer and closer to the carrier. You can also see this with daily stock market prices where the cost vs time is a random walk process (non-stationary) but the daily difference would be a stationary white Gaussian noise process (assuming no memory in the market which is typically valid for prices from one day to the next- even from one minute to the next which is why traders are concerned with  ns time intervals with regards to trading information / communications)
And I have other posts here on DSP.SE that demonstrate the application and utility of the Allan Variance and Allan Deviation beyond the evaluation of Clocks:
see
How to interpret Allan Deviation plot for gyroscope?
and
What determines the accuracy of the phase result in a DFT bin?