Is the word "that" used correctly in the following sentence? "His excuse, that he had fallen asleep on the bus, was hard to believe." What kind...

The sentence is correct, and the clause is a non-restrictive adjective clause (a type of subordinate clause).  An adjective clause begins with a relative pronoun (who, which, whose, whom, or that), as it does in your sentence, or with a relative adverb (when, where, or why).  The function of an adjective is to describe a noun, and so an adjective clause has the same function:  it describes a noun.  In this case, “that he fell asleep on the bus” complements “excuse.” 

The clause is non-restrictive because it is not essential information.  We could understand the sentence just fine if we took it out:  “His excuse was hard to believe.”  Non-restrictive clauses can always fit within parentheses, as this clause can.  In comparison, a restrictive clause provides information essential to identifying the noun in question -- good examples and explanations can be found here.