Abarim Publications' online Biblical Greek Dictionary
The noun λινον (linon) means flax. It occurs only twice in the New Testament and denotes items that are made from flax, such as a linen garment (Revelation 15:6), or the wick of a lamp (Matthew 12:20). In extra-biblical texts this word is also used to denote cords or fishing lines and subsequently fishing nets or bird hunting nets.
Even the mythical thread of destiny spun by the Fates was known by this word. One of the sons of Apollo was called Linus, and he was considered the inventor of melody and rhythm, probably because of the use of instruments with linen strings (in Hebrew Scriptures this quality is ascribed to Jubal).
In English and other languages, this noun exists as linen and is the source of words like lining and liner. In Latin, it appears to even have formed the familiar noun linea, meaning (linen) thread and hence line in general and plumb-line specifically.