The question arises, What is so distinguished about this Day that entitles it to such a special name?
Several answers come to mind:
1. There is, in fact, nothing special about the Day. It is only that CHAZAL, the Sages, in their great sensitivity were concerned even for the feelings of inanimate objects, and even for the feelings of "entities" such as units of time, like days, the existence of which is even more tenuous than that of "objects." In the case of objects, we find them concerned for the feelings of embarrassment of the challot, the Shabbat loaves, while Kiddush is recited over wine, requiring the challot to be covered. So they assigned a name to the Day, that sat forlorn in the shadow of Shavuot and the Shloshet Yemei Hagbalah, the "Day of Distinction," a name that it didn't really deserve.
2. On the First of Sivan, HaShem instructed Moshe to tell the nation, "You shall be a Kingdom of Priests and a Holy Nation" (Shemot 19:6). On the Second Day of Sivan, Moshe delivered this message and the People answered, "Everything that G-d has commanded we will do" (Shemot 19:8). So that great Day of Acceptance of the Torah by the People was given the title "Day of Distinction."
3. This Day symbolizes the machloket, the disagreement involving the Tanna Rav Yosi, an example of the Oral Torah, the Torah She-B'al Peh, that is the foundation of our belief and practice. It is the existence of this Day in this Week that makes this machloket possible.
4. Yahadut, Judaism, is fundamentally a meritocracy. A person is rewarded for his own achievements, not those of his fathers or his sons. This is exemplified by the principle that "a Torah Scholar who is a 'mamzer' (the product of an illicit sexual union), is considered superior to a Kohen Gadol, who is an ignoramus in Torah." Thus yichus or yachas, good family ties, should matter less than one's own achievements. If we take the word yachas, spelled 'Yud,' 'Chet,' 'Samech' out of the word Meyuchas, Special, we are left with 'Mem' and 'Vav,' or forty-six, as in the Forty-Sixth Day of the Omer.