This may be more effort than it's worth, but you could find a site out there that your client doesn't know about, with a suitably obscure name which does have a tagline. Grab a screenshot, and edit out the tagline, then show them that for a couple of seconds (literally, show and hide for maybe 2-3 seconds) and see if they know what the site is about. Then do the same, only this time with the tagline intact.
Get them to think about their own behaviour when they're looking for something, and get to a site they've never been to before and quickly decide it's not for them. And you might want to show them the basketball gorilla video just to highlight how little people really see of what's in front of them.
Once they understand their own behaviour they may be more willing to understand other people's behaviour with respect to their site, and more willing to understand that they need to be generous in helping people understand what they're about. The "but they ought to..." arguments will hurt their bottom line. What people should do is irrelevant - it's what people actually do that matters.
- Pro's - quicker understanding of what they do - reduced bounce-rate
- Con's - none