Why does the Torah see fit to mention the famine that occurred in the time of Avraham?
Ramban #1: Because, it appears, that had been the first famine ever; 1 so the Torah uses it as a time marker.
Ramban #2: Because people still remembered it and spoke about it; above all, they recalled how, on account of it, Avraham went down to Egypt, where he was treated royally. And that explains why Yitzchak had in mind to follow in his father's footsteps, 2 and had to be told (in 26:2) not to go down to Egypt.
Moshav Zekenim: There had already been a previous famine in the days of Yitzchak. Only this one was as harsh as the one in the days of Avraham.
Ha'amek Davar: "Milvad" implies right after the previous [event], or it pertains to it. Here, it was about 100 years later! Rather, it resembled the first famine. Initially, the entire world conducted naturally, including Eretz Kena'an. In the first famine, Hashgachah Pratis was seen - people received according to their deeds. Over time, this was forgotten. The famine in Yitzchak's time came to renew this; people saw awesome Hashgachah Pratis in Yitzchak's success.
Considering what happened to Avraham when he went to live in Eretz Pelishtim, could this have been the same Avimelech as the one with Avraham?