The wrapping below the button is not only more durable, but is actually frequently used when replacing buttons on blazers, sport coats, and the like as a protection from friction as the garment moves and shifts while being worn.
You can use common cotton thread found in most basic sewing kits, but for a stronger, more durable repair you might consider "button thread". It is a cotton-blend thread that is a bit thicker and stronger due to the polyester content.
It can be found in any good craft or fabric shop and is only about 50 cents more than the average spool of common thread.
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