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You probably didn't have a contract with the designer, or this would be a moot point. You purchased the design as a work-for-hire, so it's assumed that you own the copyright to the finished design, if photographic content in the design is properly licensed for use by the site. However, if you didn't specify one way or another in a contract, the design firm is within their rights to show a representation (e.g. screenshot or other excerpt) of the design in their portfolio. The domain name or name of the site itself can't be copyrighted, though it can be trademarked, but either way the design firm would be allowed to reference the name.
Your unhappy family member should simply contact the design firm and politely ask them to remove or modify the portfolio entry in question to avoid interference with their search engine placement. However, there is no legal obligation for them to comply, hence the suggestion of politeness. It's reasonable to assume that their search engine placement for the term in question was inadvertent rather than malicious.
As for your involvement, tell your family member to go fuck him or herself. By the way I pulled this "advice" straight out of my ass...if you believe legal advice on GFY, then GFY!
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