(120b) "v'Avarti b'Eretz Mitzrayim... Ad Chatzos..." Makos Bechoros was exactly at Chatzos. Why is the Mitzvah more from night until Chatzos, than from Chatzos until morning?
Both opinions (Rebbi Eliezer and Rebbi Akiva) agree that the time that the Mitzvah begins is at nightfall; everyone agrees that this is the intent of the verse, "k'Vo ha'Shemesh," as explained in Berachos (9a). The only argument is when the end of the time of the Mitzvah is, for this is not written clearly in the verses. Since the Torah commands, "v'Achlu ba'Laylah ha'Zeh," it must be referring to the end of the time of eating it.
Rebbi Elazer ben Azaryah is learning the obligation to eat the Korban Pesach from the killing of the Bechoros with a Gezeirah Shavah. If we view the eating of the Pesach that year as a protection against their own inclusion in the Makah (as a sort of Tikun), then it makes sense to say that they had to eat it from the beginning of nightfall, until such time as the danger of being included was over; i.e. until midnight, when the plague was finished, and they could safely stop, since they no longer needed to be protected.
Lovely to hear from you. A Lichtigen Chanukah, Kol tuv