My son uses a palate expander and the side prong has broken off. Since it's so late I can't call the dentist but my son is very upset because he doesn't want to have to get a new one or prolong the process.
Is the side prong really necessary?
Ya of-course, the side prong is the core working piece in buccal cavity expander/palate expander, without it you can't obtain the purpose of installing the palate expander.
I think what you mention is some kind of dental brace (Google search result) and I edited your question a little.
I'm not a dentist, but these things are shaped that way for a reason. It follows that its components are all necessary, or else they wouldn't be there. After all, this is not a toy or a gadget that would benefit from any artistic additions.
Don't panic because it's weekend, but make an appointment early the following week. They usually make room for "emergencies" like this (as opposed to planned, routine appointments).
Whatever it was that fell off, needs to be replaced. "Prolonging the process" is more likely if you ignore this because the part has purpose that is no longer performed.