Website redesigns can be a double-edged sword for SEO - done incorrectly, the results could mean disastrous consequences including significant drops in organic traffic. However, when done properly it has the potential to provide an added boost of authority and improved UX that ultimately leads to higher rankings.
They can:
improve SEO and site performance
simplify the user experience
refocus your content strategy
give your site a facelift
offer faster load speeds
provide up-to-date security