Abaye was a Kohen and was eligible to receive the coveted Zro'a, L'Chayayin, and Keiva, shoulder, tongue, and stomach of every animal slaughtered. The gemara Chulin (133a) tells us that initially, Abaye in his enthusiasm to show how important the mitzva is, used to grab these pieces of meat from the people who slaughtered the animal. Later when he heard that the pasuk says these pieces should be "given" to the Kohen, he stopped taking them himself but started to tell the people to give it to him. When he heard that the Navi criticized the children of Shmuel HaNavi for "asking" for the Matanos, he stopped asking but continued to accept them when offered. When he heard the Braisa says that the modest Kohanim would pass on the opportunity to get a piece of the holy Lechem HaPanim while the aggressive ones would grab, he stopped accepting altogether.
The Mei Shiloach says that the nature of a person whose opinion is attacked or even questioned, is to stand up and defend himself vigorously. This is especially of a person of stature and even more so when it comes to his personal conduct. Admitting error puts a blemish on his past behavior, which a public persona has trouble dealing with both on his own account and that of his position.
Abaye exhibited the exact opposite behavior. Despite that after his own internal lengthy debate, he decided that grabbing the Matanos showed the most respect for the Mitzvos, as soon as he even "heard" that his way may not be correct, rather than defend himself he chose to change his ways. Still when that did not prove sufficient to stem the voices of dissent, Abaye once again altered his behavior without any argument. And then he did it for third time. Could you imagine the shame of a Gadol HaDor swallowing his pride three times over the same issue?
Abaye, says the Mei Shiloach, set an example how all of ones conduct must be totally L'Shem Shamayim without any consideration of ones own ego. One must always seek the truth no matter what is at stake for him personally.