There are only 4 instances of kə·sūṯ (Strong's H3682) in the Old Testament: Genesis 20:16, Job 24:7, Job 26:6 and Job 31:19, each of which is associated with "covering" nakedness. Adding to that the "thou shalt not uncover the nakedness of" expression of the KJV in Leviticus 18, becoming "you shall not have sexual relations with" in the NIV, there can be little doubt what Abimelech's compensation to Abraham was all about.
I'm left wondering, however, whether the tranlators have actually rendered the pronoun correctly in this verse. Perhaps it would have been better translated:
"And to Sarah he said, Behold, I have given your brother a thousand pieces of silver: behold, IT is to you a covering of the eyes, to all that are with you, and with all other: thus she was reproved."
I'm not a Hebrew scholar by any means, but I find it hard to see how Abraham, in any way, becomes a "covering of the eyes" for Sarah.
The passage from which Genesis 20:16 belongs, clearly portrays a public gathering: Abimelech calls his officials (Genesis 20:8), and then he summons Abraham (Genesis 20:9), and in front of these people Abimelech humbles himself in the sight of the Lord.
In his explanation to Abimelech about why he had behaved as he had, Abraham says, "There is surely no fear of God in this place;". However, this was clearly not true, which is evidenced by:
Abimelech's appeal to God - "Lord, wilt thou slay also a righteous nation?". The word "also", here, suggests that Abimelech had heard of God's intervention to destroy the wicked nations of the plain, i.e. Sodom and Gomorrah, and his people were not like theirs.
Abimelech's behaviour - he did not wrongfully look on Sarah. His actions were with integrity of heart and innocence of hands (Genesis 20:5), which God, himself, confirms as true (Genesis 20:6), and as much as for Abimelech's righteous behaviour as for Sarah's honour, God made sure nothing improper took place.
Abimelech's acknowledgement of Abraham as God's prophet, whose prayer would be needed to redeem his life (Genesis 20:7).
The reaction of Abimelech's officials - "when he told them all that had happened, they were very much afraid" (Genesis 20:8)
Abraham's assumption was wrong about there being "no fear of God in this place", and the trust he put in his own insight caused a major issue for himself and the people of the land. Besides being an important lesson for Abraham, God used this encounter as a means of advertising His presence, and His willingness to intercede on behalf of the righteous as well as against the unrighteous.
Conclusion
The willingness of a man of power and substance, such as Abimelech, to humble himself before men, in the sight of "those with" Sarah (Abraham's people) and "all other" (Abimelech's people, and anyone else who would hear of the incident - including us), and thus before God, by offering so great a compensation, was surely the proof that Sarah's nakedness had not been uncovered.
Sarah was certainly vindicated by Abimelech's righteous intent and humble behaviour, but Abraham and Sarah were also reproved/rebuked for their dependence on themselves rather than God. So, regardless of the writer's intent for the Hebrew word וְנֹכָֽחַת׃ (wə·nō·ḵā·ḥaṯ - Strong's H3198), I think the meaning of the passage is pretty clear, and either meaning for this difficult word is acceptable.
