I am using the Canvas WordPress theme (by WooThemes.com) and successfully made a child-theme to override my CSS. However, I would also like to apply the same child-theme principles to my theme's "includes" folder which contains a variety of .php files. Any idea how I can go about doing this?
Depending on how Canvas calls in those files, you might not be able to do this.
If they're using locate template or something similar (get_template_part, for instance), you'll just need to create your own includes folder in the child theme and name the files the same as their couterparts in the parent theme.
If the parent theme uses require or require_once (or include/include_once), there's not going to be an easy way to override those parent theme files.
What are you trying to replace?