I have read about that but my project is going to run from battery with a few minutes of sleep mode, then wake up and get the readings and so on.
If I understand, in this case I couldn't use smoothing.
I agree that a running average is not for you in this case.
You're gonna deal with two types of noise here: noise generate by the sensor and noise generated by wind fluctuations. I have no experience in measuring wind, but my intuition is that the latter will be so much more present that you don't care about the former.
With that established, you need to think about how long you will need to measure in order to get a representative sample. Think about what is the lowest frequency fluctuation you want to filter out, and make sure you measure at least for a few periods. Is the wind making the vane oscillate at 1 Hz (=1 period/second), and you want to average that out, make sure you measure for at least 5 seconds, and take at least 50 samples in that period. If you then take a arithmetic mean, you will have a pretty representative sample.
So in short: use an arithmetic mean, but consider timescales for determining how long and often you measure.