You could redefine "prime" to include 1 if you wanted to without changing the fundamental truth of important results about the natural numbers. It would just mean you'd have to phrase many of them in a slightly more roundabout way, e.g. with "suppose x is a prime number not equal to 1" needing to appear in key theorems where before "suppose x is prime" sufficed. (Perhaps some other theorems might be more concise, although I'd be surprised if there were a net benefit -- the community has probably had this debate a bunch of times before settling on the current consensus.)
Definitions are chosen to be convenient for the purposes of communication, there's nothing inherently true or false about them. An argument for one definition being better than another might involve demonstrating that mathematical results people care about can be stated more clearly, concisely and efficiently using your alternative definition. The recent tau vs pi controversy providing an amusing example :)