Gemara Shabbos 20a and in other locations mentions כמאכל בן דרוסאי Rashi explains that Ben Drosai was a thief who ate his food when it was only cooked 1/3. Did Ben Drosai have some sort of Zechus that his name is often mentioned when it comes to Halchaos of Bishul on Shabbos?
According to this article, Rashi assumes that he was a thief, based on logic and not tradition. Based on the Yerushami, he was a person who aided Rav Yochanan to remove idolatry.
According to Rabbi Dr. Shmuel Klein in his essay "לחקר השמות והכינוים", Leshonenu 1:4, ben Drosai was a Tanna named R' Yonatan ben Dores, also known as Abba Hadros (called in the Sifrei Abba Doresh (here and here), which Rabbi Klein said is a scribal error, and indeed, his correction does appear in at least one Sifrei MS brought in Finkelstein's edition - spelled אבה הדרוס in one place and אבה הדורס in the other). R' Yonatan ben Dores appears in Sifrei Zuta Bamidbar 35:12. According to Rabbi Klein, R' Yonatan joined the zealots during the Great Revolt and may be identified as the Yonatan from Wars of the Jews 6:2:10 who "went out at the high priest John's monument, and uttered many other insolent things to the Romans, and challenged the best of them all to a single combat" and when "... one whose name was Pudens...ran out to him, and was too hard for him in other respects, but was betrayed by his ill fortune; for he fell down, and as he was down, Jonathan came running to him, and cut his throat, and then, standing upon his dead body, he brandished his sword, bloody as it was, and shook his shield with his left hand, and made many acclamations to the Roman army, and exulted over the dead man, and jested upon the Romans...". As Rabbi Klein explained, Yonatan (or R' Yonatan?)'s standing on atop the fallen Roman soldier was him stomping him, דרס אותו. He probably did this per the verse "And you shall tread on their backs" (Devarim 33:29), which Onkelos translates as "וְאַתְּ עַל פְּרִיקַת צַוְּארֵי מַלְכֵיהוֹן תִּדְרֹךְ" - You shall tread on the necks of kings", and the Targum Yerushalmi translates as "עַל פִּירְקַת צַוְורֵי מַלְכֵיהוֹן תְּהוֹן דַרְסִין" - same root of דרס is used.
Therefore, it seems that this R' Yonatan was given the nickname "Abba Hadros" or "ben Dores" by the people after his exploits during the Revolt (where he was also killed, according to Josephus). The title "Abba ___" is similar to "Abba Sikra", leader of the zealots in Yerushalayim, which comes from "sicarii" and "Abba Techina Chasida" who according to Rabbi Klein, was called Techina ben Perisha and later became known as "ben Haratzchan" (the murderer)1.
This R' Yonatan was probably a vicious man, and to show his lust for the blood of his enemies, would eat his food cooked only a third of the way, and as an echo of his legend, such a meal became known as a "ben Drosai meal".2
1 Elazar ben Dinai mentioned with ben Haratzchan is identified by Rabbi Klein in his book Ever Hayarden Hayehudi, pg. 22 as the Elazar who together with Amram, led an East-Bank rebellion against the Romans (Antiquities 20:1:1) and this Amram can be identified with the Amram mentioned in Shir Hashirim Rabbah 2:7.
2 Somewhat ironically, if this interpretation of the name and story of ben Drosai is correct, then the common mispronunciation of his name as "ben Dorsai" also fits...