The second day of Sivan is called “Yom Hameyuches, the day with lineage.” There are various reasons given for this moniker.
Some say that though it has nothing special occurring on it, it is honored because the day before is Rosh Chodesh and the next day begins the Shloshes Yemei Hagbala, the three days prior to the giving of the Torah at Sinai when the Jews prepared themselves to receive it. By virtue of its “relatives,” it is a special day.
Some say it’s precisely because it’s surrounded with special days and it is not, that teaches us humility which is worthy of honoring.
Another reason is that this is the day when Hashem said He wanted us to become a kingdom of Priests and a holy nation. When we said Naaseh v’Nishma then, we sanctified the day.
The Rambam grapples with the concept that Hashem held Mount Sinai over our heads to force us to accept this. How is that binding?
He answers that we descend from our holy Patriarchs, who instilled in us the desire to serve Hashem. We want to do it but sometimes need a push.
When Hashem held the mountain over our heads, it merely reinforced the willingness we had from our lofty ancestors, hence this is the day of lineage.
Based on a shiur by Heshy Hollander