of is a preposition, and prepositions must be followed by a noun. With many verbs, you can make a noun-form (called a gerund) by adding -ing to the verb. Generally we only do this when there isn't already a suitable noun. English people like to talk about rain, so there are many suitable nouns, and at least two are suitable: rain and rainfall. It therefore isn't necessary (and sounds wrong) to use the gerund of heavy raining.
We use a before a singular, countable noun to indicate that we are referring to a non-specific whatever-it-is.
For non-countable nouns like rain (note the U after noun in this definition), we don't normally put a in front of it:
it is the sound of heavy rain
The exception is if we wish to refer to a non-specific example of a particular type of something. Here is another example where a non-count noun is qualified by an adjective:
The colour of old claret
The colour of an old claret
The first sentence suggests that there are many types of claret, that old claret is one of these types, and that it has a particular colour. The second sentence further suggests that there are many types of old claret, but any non-specific one of them has (within reasonable limits) this particular colour.
In your sentence, a heavy rain suggests that there are many types of heavy rain, but they all have (within reasonable limits) the same sound. While this meaning is possible, it is, in my opinion, unlikely.
storm is countable (note the C in this definition), so for a non-specific storm you would always include an indefinite article a:
it is the sound of a heavy storm