How do I go about producing a new product as a give a way business promotion? Like the live strong armbands but different?

There are literally dozens of websites that offer customized merch for just such things.
Just google “custom promotional items” and related things to find lots and lots

If it’s a totally new product, one which has never existed before, you’ll have to get it manufactured to your specs. Usually, you’ll need to have it manufactured in a large quantity to make it worthwhile monetarily. If this is what you’re thinking of, I strongly suggest you rethink it. Why reinvent the wheel for a freebie item?

@augustlan, there’s money to be made in ad specialty items.

@jegonza89, you have to first patent the item, then work with a manufacturer to get the item produced in prototype quantities. They you need to take the prototypes to ad specialty conventions and do sales to ad specialty companies. That’s where you would take orders. They are the middle men between your product and the company that would use the item as a give away to promote their business.

@AlfredaPrufrock I think I was looking at this bass-ackwards. I assumed he meant that his company wanted to give them away, not that he wanted to manufacture the item to sell to other companies. Your point makes perfect sense.