Why does the Torah see fit to mention the famine that occurred in the time of Avraham?
Ramban #1: Because, apparently it was 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: Because there had already been an earlier ? less severe famine in the days of Yitzchak, and this one was as harsh as the one in the days of Avraham.
Ha'amek Davar: Because usually, "Milvad" implies right after the previous [event], or it pertains to it, whereas here, it was about a hundred years later ? to teach us that it resembled the first famine. Initially, the entire world conducted naturally, including Eretz Cana'an. In the first famine, Hashgachah P?ratis 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.
Was this the same king who took Sarah in the time of Avraham?