The phrases חדש החשבון, חדש הרחמים, and חדש אלול refer to different aspects of the month of Elul, which is a time of introspection, repentance, and preparation for the High Holidays.
This refers to Elul as a month when individuals are meant to engage in a personal reckoning or accounting of their deeds and spiritual state over the past year. The word חֶשְׁבּוֹן (cheshbon) means "calculation" or "account," and this period encourages introspection and self-evaluation, leading to תשובה (repentance).
Elul is often called the "Month of Mercy" because it is a time when Hashem's compassion is especially accessible. According to tradition, during Elul, Hashem is more receptive to prayers, forgiveness, and the repentance of His people, preparing for the Days of Awe—Rosh Hashanah and Yom Kippur. This is based on the verse from Shemot (Exodus) 34:6-7, where the י"ג מידות הרחמים (13 Attributes of Mercy) are emphasized during the Selichot prayers.
Elul itself is a special time of preparation for the High Holidays, marked by increased prayers, the sounding of the שופר (shofar), and the practice of סליחות (penitential prayers). The month is seen as a unique opportunity to strengthen one’s relationship with Hashem and return to a path of righteousness.
Together, these phrases highlight the multifaceted spiritual themes of Elul: personal reflection (reckoning), seeking mercy, and preparing for renewal and change as the Jewish New Year approaches.