Questions & Answers about Duono de la ŝtofo estas kotono.
Duono is a noun meaning half. The ending -o marks it as a noun.
In this sentence, duono de la ŝtofo means half of the cloth/fabric, and that whole phrase is the subject.
By contrast, duon- often appears as a root in compounds, such as duonhoro (half an hour), or in forms like duone (halfway / partly).
This is a very common question.
Use de when you mean a part of a specific whole:
duono de la ŝtofo = half of the cloth
Use da when you mean an amount or quantity of something, usually an unspecified substance:
kilo da sukero = a kilo of sugar
So here, de is right because the sentence is talking about half of a particular piece of cloth/fabric, not just some quantity of cloth in general.