Answer: The subordinate clause is "When I told you that I didn't want any tuna salad," and it is used as an adverb.
Step-by-step explanation: While "I was quite serious" can be a sentence in itself," the clause "When I told you I didn't want any tuna salad," doesn't, because it is subordinate to "I was quite serious," as the subordinating conjunction "when" implies. This subordinate clause is clearly used as an adverb, because it modifies the verb "was."