Note: the following explanation is not necessarily what I believe personally. However I think it is important to include it as a possible answer since it is based on an important school of biblical criticism:
The documentary hypothesis provides a hermeneutical approach that yields a simple answer: Sarah was not actually as old as she appears, because the narrative should not be taken as strictly chronological. In fact, according to this hypothesis, the narrative of Sarah and Abimelech is a reiteration of the story of Sarah and the Pharaoh in Gen. 12:11, where she is described as "a beautiful woman." But meanwhile the earlier story has unfortunately already established her as old and "worn out." (18:20) A footnote in the NABRE translation explains:
Abraham again passes off his wife Sarah as his sister to escape trouble in a foreign land (cf. 12:10–13:1, the J source)The story appears to be from a different source (according to some, E) and deals with the ethical questions of the incident.
The hypothesis is based in part on the fact that Gen. 12 uses the name yhwh (the Lord) for God while Gen. 18 uses the word elohim (God/gods). These are general markers for the J source (j standing for jehovah/yahweh) and the E source (elohim). In addition, Abimelech is described as a Philistine (Genesis 26:1), but they did not arrive in the area until about 800 years later, lending support to the idea that the story evolved over time.
The theory serves to explain various doublings, differences in names (Abram/Abraham etc.) and other apparent contradictions in the text. Sarah may still be old by modern standards, but not so old as to prevent her from being very attractive to men.
Conclusion: according the documentary hypothesis, Sarah was not really as old as she appears. What we have in the text is two versions of the same story, in which Sarah is extremely attractive. The biblical editor, rather than omitting one sacred legend in favor of the other, included both - with one occurring much later on. What attracted Abimelech to her was her great beauty.