We just read that the people grieved after Aharon much longer than after Moses and the Midrash infers that Aharon was involved in bringing peace between people and especially couples.
When and how was Aharon implementing his peace-making strategy - did he reach out, or people come to his place, did he have a special Beis Midrash people could attend, and when did he have time if he was learning Torah with Moses when he didn't serve in the Mishkan?
This is the commentary of Korban Haedah on Yerushalmi, Sanhedrin 1:1:
Talmud:
תני ר׳ אליעזר בן ר׳ יוסי אומר המבצע חוטא והמברך את המבצע הרי זה כמנאץ לפני המקום שנאמר ובוצע ברך נאץ ה׳ אלא יקוב הדין את ההר כשעשה משה אבל אהרן משים שלום שנאמר בשלום ובמישור הלך אתי.
Commentary:
אבל אהרן: כששמע מחלוקות ביניהם, קודם שבאו לפניו לדין היה רודף אחריהן ומטיל שלום ביניהם.
When Aharon would hear an argument taking place, he would run after them before they even came to court, in order to make peace among them.
Aharon would go to them. See Avos Derebbi Nassan:
וכן שני בני אדם שעשו מריבה זה עם זה הלך אהרן וישב אצל אחד מהם אמר לו בני ראה חברך מהו אומר מטרף את לבו וקורע את בגדיו אומר אוי לי היאך אשא את עיני ואראה את חברי בושתי הימנו שאני הוא שסרחתי עליו הוא יושב אצלו עד שמסיר קנאה מלבו. והולך אהרן ויושב אצל האחר וא״ל בני ראה חברך מהו אומר מטרף את לבו וקורע את בגדיו ואומר אוי לי היאך אשא את עיניו ואראה את חברי בושתי הימנו שאני הוא שסרחתי עליו הוא יושב אצלו עד שמסיר קנאה מלבו. וכשנפגשו זה בזה גפפו ונשקו זה לזה
Similarly, when two people were fighting with one another, Aaron would go and sit next to one of them and say: My son, look at the anguish your friend is going through! His heart is ripped apart and he is tearing at his clothes. He is saying, How can I face my old friend? I am so ashamed, I betrayed his trust. Aaron would sit with him until his rage subsided. Then Aaron would go to the other person in the fight and say: My son, look at the anguish your friend is going through! His heart is ripped apart and he is tearing at his clothes. He is saying, How can I face my old friend? I am so ashamed, I betrayed his trust. Aaron would sit with him until his rage subsided. When the two people saw each other, they would embrace and kiss one another.
In a similar vein to the other two answers, the Mishna in Pirkei Avos (Ethics of the Fathers) 1:12 famously writes:
הִלֵּל וְשַׁמַּאי קִבְּלוּ מֵהֶם. הִלֵּל אוֹמֵר, הֱוֵי מִתַּלְמִידָיו שֶׁל אַהֲרֹן, אוֹהֵב שָׁלוֹם וְרוֹדֵף שָׁלוֹם, אוֹהֵב אֶת הַבְּרִיּוֹת וּמְקָרְבָן לַתּוֹרָה:
Hillel and Shammai received [the oral tradition] from them. Hillel used to say: be of the disciples of Aharon, loving peace and pursuing peace, loving mankind and drawing them close to the Torah.
Rashi then proceeds to give a detailed description as to the process Aharon took. He writes as follows:
כשהיה רואה שנים מריבים זה את זה. היה הולך לכל אחד ואחד שלא מדעת חברו ואומר לו: פלוני, ראה, למה אתה מריב עם פלוני? הלוא הוא נשתטח לפני ופיסני לבוא אצלך לדבר עמך עליו, עד שתתרצה להתפייס עמו. וכמו כן היה מדבר לחברו. ומתוך דברים הללו היה משים שלום ביניהם. אמר אחד לאשתו: קונם שאין נהנית לי עד שתרק בעינו של כהן גדול. היה אהרן שומע, היה הולך אצל אותה אשה והיה אומר: פלוניתא, חשתי בעיני, ורק שלך רפואה יש בו. תרק לי בעין. והיתה רוקקת. ועל כן נאמר בו (במדבר כ:כט) "ויבכו את אהרן שלשים יום כל בית ישראל" - אחד אנשים ואחד נשים במשמע, ובמשה לא נאמר אלא "בני ישראל", כדכתיב (דברים לד:ח) ויבכו בני ישראל את משה
When he saw two people fighting this one and that one, he would go to each one without each one's friend knowing and say to him: "Ploni - see, why are you fighting with ploni? Did he not come and prostrate himself before me and appease me to come to you to speak with you about him, until you will want to reconcile with him". And similarly he would speak to the other friend. And through these words, he created peace between them. One would say to his wife: "You won't benefit from me until you spit in the eye of a high priest." Aharon would hear and he would go by the woman and say: "Plonisa - I am worried about my eye and only you have the cure/medicine for it - spit in my eye." And she would spit. And therefore, that's why it says: "All of the house of Israel cried about Aharon for thirty days" (Bamidbar 20:29), implying both the men and the women. Whereas by Moshe it only says the "Children (Bnei) of Israel like it writes: "And the children of Israel cried...for thirty days" (Devarim 34:8)
Note the Kallah Rabbasi 3 which aside from bringing a similar scenario listed by Rashi also adds about how Aharon would intervene with quarrelling spouses:
When [Aaron] heard of a husband and wife who had quarrelled, he would go to the husband and say to him, ‘[I have come] because I heard that you have quarrelled with your wife; should you divorce her it is doubtful whether you will find another like her or not; and further, should you find another and quarrel with her, the first thing she will say to you will be, “You must have behaved in a like manner towards your first wife” ’. In consequence of this all Israel, men and women, loved him.
Indeed, such was his success with making peace amongst husband and wives it is worth noting parenthetically (a few lines on in the Kallah Rabbasi):
Eighty thousand young men named Aaron followed Aaron’s bier. They were the issue of those who wanted to divorce but retracted and their wives became pregnant.
I.e. they named their resulting sons Aharon out of gratitude to him for saving their marriages!!
