"There was a famine in the land, besides the first famine that was in the days of Abraham. And Isaac went to Abimelech king of the Philstines, in Gerar. Then the LORD appeared to him and said: 'Do not go down to Egypt; live in the land of which I shall tell you. Dwell in this land, and I will be with you and bless you; for to you and your descendants I give all these lands, and I will perform the oath which I swore to Abraham your father. And I will make your descendants multiply as the stars of heaven; I will give to your descendants all these lands; and in your seed all the nations of the earth shall be blessed; because Abraham obeyed my voice and kept My charge, My commandments, My statutes, and My laws.' So Isaac dwelt in Gerar" (Gen. 26:1-6).
About 100 years earlier, there was an extreme famine in the land and Abram went to Egypt for sustenance (cf. 12:10). Apparently, Isaac was headed for Egypt also, but was stopped by God first and instructed to remain in Canaan. God encouraged Isaac by reiterating the Abrahamic covenant with him (i.e., specifically the nation, land, and seed promises). God promised to bless Isaac if he dwelt in the land. He would give all of Canaan to his family, multiplying his descendants beyond measure. Through Isaac's lineage would all of humanity be blessed. There was no need for Isaac to be concerned about a famine--God was watching over him! Isaac stayed in Gerar where Philistines lived.
"And the men of the place asked about his wife. And he said, 'She is my sister'; for he was afraid to say, 'She is my wife,' because he thought, 'lest the men of the place kill me for Rebekah, because she is beautiful to behold.' Now it came to pass, when he had been there a long time, that Abimelech king of the Philistines looked through a window, and saw, and there was Isaac, showing endearment to Rebekah his wife. Then Abimelech called Isaac and said, 'Quite obviously she is your wife; so how could you say, "She is my sister"?' Isaac said to him, 'Because I said, "Lest I die on account of her."' And Abimelech said, 'What is this you have done to us? One of the people might soon have lain with your wife, and you would have brought guilt on us.' So Abimelech charged all his people, saying, 'He who touches this man or his wife shall surely be put to death'" (Gen. 26:7-11).
Is this a case of a father's sins influencing a son? Perhaps, although both records of Abraham lying were before Isaac's birth. More likely, Isaac and Abraham simply thought the same way, and there was good reason (in their minds, at least) for not wanting to be perceived as being married to gorgeous women as strangers in another land. Whatever risk they thought they were taking by telling the truth, their lies are not justified. Honesty was (and is) the proper course. God had made great promises to these men; why should they fear for their lives because of their beautiful wives?
The mentioning of Abimelech again is ambiguous (cf. 20:2). Is this the same man Abraham lied to? Such is possible but not demanded by the text. "Abimelech" is not a name but a title (e.g., President or Pharaoh are also titles). So, this could have been the same man, but we cannot know for certain. Abraham lied to an Abimelech approximately 80 years earlier. Abraham lived to be 175 years old, so had Abimelech been young when he took Sarah into his harem, this could be the same man--though an old man now. The fact that he was interested in Sarah sensually does suggest he was young at that time. The Abimelech Isaac is dealing with is likely much older, for he sees beautiful Rebekah but does not have an interest in her for himself (cf. 26:10). Abimelech sees Isaac "showing endearment" to Rebekah; clearly, the two are married! Abimelech confronts Isaac and rightly rebukes him. Abimelech orders that none of the people touch either Isaac or Rebekah. The fact that Abimelech understood (as did the Abimelech that Abraham lied to) that it would have been a sin to have intimate relations with a married woman conveys something about the morality of the Philistines at this time. They have not completely abandoned Jehovah's ways yet (cf. 15:16).