The identification of the "poor wise man" in Qohelet (= Ecclestiastes) 9.14-16 (who rescued his city from the siege) with Archimedes (who rescued Syracuse from several Roman sieges) goes back to Moritz Friedlander in his book Griechische Philosophie im Alten Testament, Berlin, Reimer 1904, pp. 153-6 (Reprinted Vero Verlag, 2014). There is much to say in favor of this identification, because it matches in many details Qohelet's text, but this needs a long discussion. Peter writes: "In Ecclesiastes 9:15 the little city was delivered. This was not the case for Syracuse." But Polybius tells that Archimedes rescued Syracuse continuously from many Roman attacks by his clever mechanical devices, and this is what Qohelet means (although the city fell at last and Archimedes was killed by a Roman soldier).
In a paper "Ecclesiastes" by Paul Haupt (The American Journal of Philology, Vol. 26, No. 2 (1905), pp. 125-171) the author argues convincingly (p. 126) that
Qohelet (=Ecclestiastes) must have heard about Antiochus IV Epiphanes (175-164 BCE); for in 4.I3-I6 he says that he saw the general enthusiasm for the poor but wise youth (i. e., Demetrius I Soter or Alexander Balas (a pretender)) who succeeded (I50 B. C.) to the throne of the old and foolish king (i. e., Antiochus IV Epiphanes who was described as a madman). There is much to say in favor of this identification as well, but this needs a long historical and philological discussion.
My guess is that Qohelet is one of the earlier Hasmonean priest-kings (perhaps John Hyrcanus I (134 BCE) or Alexander Jannaeus (100 BCE)), because his "water pools" described in Qohelet 2.6 match the water pools of the Winter Palaces of the Hasmonean kings near Jericho, excavated by the archaeologist Ehud Netzer in the mid 1970's.