Work on a combination of easy access (speed of the site loading), intuitive navigation...and relevant/dynamic content with the appropriate embedded links to lead deeper into the site. People need a reason to stay longer. WIthout seeing the site in question it is hard to provide more than a few generalizations.

As for design, make sure your design does not impede traffic in any way. Design the site with highly visible hooks (content to draw them in) and themes to draw them back. People often view a website in the same way they scan a newspaper. It's quick...unless something catches their eye quickly and prominently.

One of the things I would check is how your client knows the average stay is 30 seconds or less. While her stats program may be providing that information it is not always accurate or relevant. Does it include search engines?

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