Toronto textile artist Linda Heron wrote me an email about the red thread. I found it very informative and would like to share it here. (abridged)
In Buddhist tradition and others too I think, the red thread is called a blessing or protection thread, and we tie one knot to remember the 3 Refuges (Buddha spirit in us all, Dharma teachings and Sangha community), a second knot to remind us of compassion for ourselves and all beings and a third one to remind ourselves of something that will benefit all, and to cultivate its growth. The red or maroon thread is symbolic of having come from a monk or nun's robe.