According to the Ibn Ezra, the eulogies were merely sad dirges that the Jews composed and sang at this time.
Yad HaMelech goes along with his earlier comment that there were places far from the capital of Shushan where the king had less ability to control his citizenry. Therefore, he maintains that the Jews were mourning those who were indeed killed in the places where persecution began, despite the king’s decree.
According to Rav Shach, who quotes Rav Isser Zalman Meltzer, the Jews were eulogizing the days wasted without Torah study and mitzvah performance. Indeed, in this world where time is all we have, such a waste truly deserves mourning.