With your needle in between the fabric and button, flip it back over to face the front, then start wrapping the thread around ...

But if you have the patience, I truly believe anyone willing to wield a needle and thread can be successful with this DIY fix ...

Once separated thread one strand through the needle. Knot the thread at one end. Start at one corner of the stocking and using a blanket stitch, sew down the side to create a decorative edge.