A commonly used term for the "that" in "They were happy that the bus came" is complementizer. It is added at the beginning of the sentence "the bus came" in order to convert it into the complement "that the bus came".
This, in turn, is the object complement of the preposition "at", which here is understood. The complement can be pronominalized to "it"/"this"/"that", in which case the "at" is explicit: "The bus came, and they were happy at that."
"that" complements also occur as the subject and object complements of verbs: "That the bus came pleased them," "They knew that the bus came." Some nouns, like "fact", can also take a "that" complement: "The fact that the bus came surprised them."