Personally, I do think the author means something like "sew" when she writes to "tease the thread through". More specifically, due to the context, I believe the woman is knitting (out of context, I would have thought she was trying to simply thread the needle).
"Here is Ada, sitting on the sofa in the good room in Broadstone. She has a piece of work in her hands, simple work, it must have been some hemming or darning. There is an eight-year-old girl in the room, who is me. I remember the curve of her back; her hands, dropped in her lap; the pick and lift of her fingers as she teases the thread through. The sofa behind her is a dark red, overlaid with a tangle of cushions, though Ada does not lean back into them."