Yes you can. However, not from your mom's sewing kit: The needle is bigger and the thread is more robust. A fabric store will have both.
Removing the seat cover is easy enough and you just rethread it through the same holes. Making your own holes can be difficult and I don't recommend it. Very difficult to get a straight line....so take it down a shop for a quick pass on their sewing machine. If the existing thread is rotting, remove with a razor blade and redo the whole section.
Drew