To paraphrase Freud, sometimes a rain shower is just a rain shower. It does not have any great symbolic significance. England has what is known as a temperate maritime climate, similar to that of Seattle or Portland, and it rains frequently.
Within the plot structure of the play, it makes a convenient way to introduce us to several of the main characters of the play in a way that is not overly contrived. In normal circumstances, a poor flower girl such as Eliza would not have the opportunity for protracted conversation with members of the upper class.
Also, for Shaw to illustrate Higgins' skill at dialects, rather than just telling us about it, we need to see him identifying accents. This needs to be done in a setting where he will encounter a random group of strangers, something the shower precipitates, as it were.
Finally, the circumstances enable Shaw to show us the amiable but ineffectual nature of Freddy.