How can a client update the content of their website?

You have a few options:

• Use a CMS like WordPress. This would ultimately be the easiest solution for every involved except you, because it would involve learning the specifics of styling your CMS. I think it would be worth pursuing this option and playing around with it.
• Teach the client how to open up the code you write in a text editor and edit content. Stress which parts they shouldn't touch.
• Develop some kind of back-end interface for your client to edit content on individual pages.
• Have your client use something like Jekyll to generate your site from raw text.

Ben is right, those are definitely right.

I can also tell you, option 2,3, and 4 the client is going to screw up or not remember how to do, and they're going to call you anyway.

Which means yes, you're probably going to have to update the site for them, for awhile, depending on your contract, or what you at least verbally agreed to.

A CMS, just simply stands for a Content Management System. That can be a variety of things, but at its root, it's a way for content on the site to be added, deleted, or modified without having to modify actual code.

Wordpress, is the largest CMS out there. You can also write your own CMS, which would be common on more advanced web service applications.

You said you don't have time to learn WordPress before the deadline, so you're in a pinch. You may be able to say that your agreement with them was the initial site/design, but that they'll need to hire someone to turn that into a Wordpress site, so they can update it themselves. Someone can write a custom theme so they can get your design, on a wordpress site. If you agreed to helping maintain this thing, than to honor your word, you're in it every week modifying it. A client may react negatively if you tell them they need to spend more money to hire someone to convert it to wordpress, so clarity will be very important here.

The nice part about wordpress though is that you can give the client a wordpress admin account with stripped away rights and privileges, so they can only change, delete, or modify content you want them to, while more important content like site or plugin settings, can not even be seen, let along modified.