In option A ..
"that twenty-one ceramic dog figurines were discovered during the excavating of a 1,000-year-old Hohokam village in Tempe, Arizona." Yes, this entire clause is the subject, and as it begins with "that" it's singular, agreeing with the verb "has." However, it's extraordinarily awkward to use such a long noun clause to begin a sentence. Furthermore, noun clauses rarely make sense as the subjects of transitive verbs, and are much more often the subjects of the verb "to be" than anything else. An example of a clear usage of a noun clause is this:
"That the problems are difficult is clear; what is interesting is how to answer them."
Both noun clauses ("that..." and "what...") are short and are the subject of the verb "to be" -- in the second part of the sentence, the first noun clause, "what...", is properly equated to another noun clause, "how...." Thus these are good sentences. The original one here, however, isn't ideal, and if there is a choice that uses a simple noun and modifiers instead of the long awkward noun clause to be the subject of "has doubled," as long as that choice makes no further errors, it will be the ideal choice.
Of the remaining choices, it's necessary to keep the subject singular, so that it agrees with the singular verb "has" that isn't underlined and therefore can't change. (B) and (D) both make the subject "figurines," which is plural, and therefore need to be eliminated.
(C) is awkward because gerunds are similar to noun clauses in that it's hard to make them logically the subject of transitive verbs -- was it really the act of "discovering" them? or rather the fact that they were discovered? -- and a noun, as in (E), would be far better. Furthermore, it's rarely correct to use a gerund after a preposition, so "at the excavating" in (C) should raise some eyebrows, while "during the excavation" in (E) is, again, far superior.
Thus (E) is the correct answer.