Depending on the client, I think there are 2 ways to approach the service side of a website.

1) Monthly Fee
2) Hourly

I prefer the monthly fee option because you can bundle in say 2-3 hours of service per month at "no charge" along with their hosting fees. I think in the long run this is more profitable especially if you have large numbers of clients.

Hourly is nice but it does not give you stability in your income.

And yep, the website itself should be considered a product and one way to make it tangible is to give the client a hard copy on CDR. Which by the way, can be a nice upsale for you. "Would you like a hard copy on CD for an extra $99."