Good advice from faster. Remember you are the professional, this is what you do for a living. You're not conning them, or trying to sell them something they dont need.
An approach would be to explain why you think it needs redoing. Have a look through the site now, and write down what you think is bad about it, in your own words..
So eg, "crap navigation, cheesy and pixelated photos, annoying hover over effect on the menu, tiny thumbnails everywhewere, 1980's colour scheme (in bad way), cant read content links on crazy colour background". Then when you have a list, categorise...
So :
Imagery : cheesy and pixelated photos, tiny thumbnails everywhewere
Navigation : crap navigation, annoying hover over effect on the menu
Colours : 1980's colour scheme (in bad way), cant read content links on crazy colour background
Then, re-write in a more professional manner...
Imagery : outdated photos, don't portray the professionalism of the company. Thumbnails are difficult to see, making it easy to miss important information links.
Navigation : Confusing navigation > too many sub levels. Hover over effect could be refined/minimised.
Colours : outdated colour scheme, accessibility concerns with link colours.
Then go grab a coffee with them, take a print out of your doc, a laptop with the website pages saved offline (nothing more embarrassing than trying to talk through a site with no internet connection), and tell them you`ve taken their initial brief, and have identified aspects of the site that fall into their specified needs, but you feel there are elements that in your professional opinion should be added to the brief to create a more professional web presence.
Remember, dont be afraid to tell them what you think they dont want to hear. Most of the time they do want to hear it, they just dont want you to mention it cos they dont want to pay for it.