What records should a CFI keep to prove they took due care in the quality of their instruction? I know there has been some cases where a flight instructor is sued after a current or former student does something that gets themselves (the student) killed or causes damage.

I know that there have been cases where a flight instructor has been acquitted by proving that they took due care in their flight instruction, and the accident was not a result of poor instruction.

What specifically would a prudent flight instructor do to ensure they are keeping detailed enough records to prove that they are providing adequate instruction and that a resulting accident would not be due to poor instruction?

What are the records a CFI is legally required to keep, and what are the records they should keep in addition to that?

This question is in context of CFIs in the United States.
Have a well defined syllabus, and then a daily activity sheet where you write down the lesson you did, the time it took, and how the student performed. If anything peculiar happens, make a note. 

Have a well-written lesson binder too. Even if you don't need to use the binder to teach a lesson (like steep turns, that's not really much of a ground lesson), have it in there anyway to show that you know what you're talking about. If you have a page that only says "Teach steep turns" and a lawyer gets a hold of it, there's nothing to prove that you actually know how to teach steep turns.

AS for legally required records:


You must sign the logbook of anyone you have given flight or ground training.
You must maintain a record of the following:

The date and name of each person whose logbook/student pilot certificate you endorse for solo flight privileges (I keep a record of every endorsement).
The date and name of each person you endorse for a knowledge/practical test, and the kind, date, and result of the test.

You must keep the records for at least 3 years.