Sheldon, thanks for your question, and
Welcome to WebmasterWorld [webmasterworld.com].
Though you are suggesting a simple redesign, with no other changes, over the years I've seldom, if ever, seen such a thing. Site changes are in fact an area where lots can go wrong, and even the simplest changes require careful preplanning, backups, and notes.
Many "purely cosmetic changes" that I've seen end up involving changes in site structure, often due to a move to a new "CMS" (Content Management System), sometimes due to addition or subtraction or reorganization of pages... all bringing about changes in internal and external linking, and/or changes in page URLs.
Even "simple" visual changes on a page can bring about major SEO changes. In almost all cases, the site designers and owners had no idea that the changes they were making would affect site SEO.
You've suggested fairly regular attention to SEO, but I'm not sure whether this suggests you've updated content, moved to https, and are mobile friendly. Is yours a large site or a small one? Are you planning to use a CMS, or is your site homegrown? Are you going to add or subtract any pages? Is your site ecommerce, or are you a publisher? Etc....
So, before we make guesses about what might change, I'm thinking it would help if we had a clearer idea about what changes you're planning to make and why.