My guess is it is an urban legend. Ground based weather radar have been used track and analyse behaviour of birds. Part of this research was done with the radar in fixed beam mode to record individual wing beat frequencies for species identification, keeping the bird in the radar beam. If the birds would be scared away by the beam this research would be almost impossible (and invalid as well). Of course, that's no hard evidence that the birds aren't scared. Also the distance between the bird and radar plays a role of course.
The radar used for that research was operating in the C-band, which is also the frequency band that most aircraft weather radars operate in. Newer aircraft weather radars operate in the X-band, which is a higher frequency band. My gut feeling is that the higher frequency is even less likely to bother birds, but then again, I am not a bird so what do I know?
jwenting suggested in a comment that this myth might have been started because of incidents where animals and humans have been harmed and killed by high power radar transmissions.
The FAA has published an Advisory Circular on recommended practices and precautions for operating weather radar on the ground. While not concerned with birds, it provides an example calculations of the output power and safe distance of a weather radar. It the example, the safe distance is less than 4.5 meter. That is too close for a bird to avoid a mid air collision.