どようびは ほんを よみます。 -> I read books on Saturdays OR I will read a book on Saturday.
は, as the topic marker, can be used to refer to Saturdays in general or the next Saturday from the time the sentence is spoken, in which case it indicates a point in time without specifically having that function.
どようびに ほんを よみます。 -> I will read a book on Saturday.
に marks the Saturday you are talking about as a specific point in time. Without context this means the next Saturday from the time this is spoken.
どようびには ほんを よみます。 -> I will read a book on Saturday.
When には is used, は directs the listeners attention to the specific time indicated by に as opposed to simply stating a specific time. It is a matter of the feeling/nuance delivered to the listener. In this case, it sets Saturday apart from other days of the week and (according to a native speaker※) gives the listener the impression that this content is something that you haven't been doing thus far. So the entire meaning rendered in English would be something like
"I will read a book on Saturday which is something I have not been doing thus far and may or may not do again."
Also, remember that the option of どようび、ほんを よみます is an omission of particles, making it more vague as it leaves the interpretation up to the listener. In a general context this usually results in a straight transmission of information, which is probably why it is used so much.
※I think it is safe to say that different people would get different impressions depending on the context of the conversation. My goal in explaining what a native speaker told me was to give an idea of the kind of nuance that can be conveyed and not to say that this is the only meaning associated with this usage.
Also, in reality there are myriad ways to get は by itself to refer to a specific point in time. For example, こんどの どようびは ほんを よみます。 -> I will read a book this Saturday. And as with any structure, に、は、and には can usually all fit grammatically and all three will convey the correct information. The difference is in the feeling and nuance conveyed.
Here is a Japanese website for Japanese people studying to be Japanese language teachers that I based part of my answer on.