Going to Bora Bora is pretty expensive and there's really no cheapest time to visit Bora Bora.
But the best times to visit Bora Bora are the months between November and April because those months have the best nice weather that will allow you to have a great time in Bora Bora.
Some people prefer to visit Bora Bora between May through October which is usually when the tourists are really crowding Bora Bora so I would recommend going to Bora Bora around November through April so that you can have a good vacation without too many people.