We have recently had direct experience of this when we were asked to rename a client’s website domain name at the end of June 2011.
As expected, for the first few weeks, the total volume of visits dropped through the floor, but as we took all the usual steps to ensure as many in links as possible were renamed, the sitemaps were correct and the domain forwarding codes were working correctly, we saw a steady rise in traffic levels, getting very close to what they were before the change after 8 weeks.
Most important of all was ensuring that the deep links resolved properly to their new equivalents. If necessary you may need to hand code every single link, but hopefully this can be automated with just one line of code on an htaccess file.
Today (almost exactly two months on) the Google Page rank of the new site’s homepage jumped back up to the same as it was on the old site. Page rank is a good general indicator that things are going OK, so this was immensely gratifying.
So, the answer to the question “how does renaming a website affect visitor numbers?” is that if you do everything you are supposed to do as part of a renaming exercise, the drop in traffic should be nothing more than a temporary blip lasting approximately 8-12 weeks. Our advice is to have a strategy in place to deal with technical and SEO issues prior to the change in order to ensure your new domain name recovers position as quickly as possible.