That's surprisingly less of a problem than I naively thought it would be. I'm upfront with clients prior to, during, and after engagements, and I try to set expectations accurately. That said, for the overwhelming majority of my clients, if e.g. the entire budget for my wedding was spent on a project whose final conclusion was "Experimental results: we now know one more thing that didn't work" that would not be a catastrophic loss.
Businesses routinely spend $$$, $$$$$, and $$$$$$$$$ on things that don't end up working out the way they were planned. They're pretty much OK with that, since they run "portfolio strategies" in terms of directions. Nobody is betting the company on a two-week engagement with me, any more than they bet the company on any particular man-month of engineering time.
n.b. This is partially influenced by being picky about clients and working with people who are clueful and have budget. This is one reason why if a startup with $X0,000 in the bank comes to me and asks to roll the dice on an engagement I give them a link to my blog, some suggestions, and invite them to come back when we're a better positioned to help each other out.
The glass half empty is that there was no benefit to the redesign.
The glass half full is that you have a more visually pleasing, refreshed web site that is at least as effective as the old design. Patrick also notes that the users tend to associate visual changes with programatic improvements, even when there are no programatic improvements (placebo effect).
