One rarely knows through what influence a word has been created.
Possibly some little Alfrida of the fifteenth century at her mother’s knee could not rightly say “linen,” and the mother and father, thereafter humoring the child, also said linsey.
No one knows.
But, if we add to the fanciful story, we may then also assume that the father, weaving a textile of flax and wool, went a step further in childish speech and called his weave linsey-woolsey.