This is a self Q&A.
How do you modify the text/html that appears in the output of a wp_nav_menu? For example, I wanted to add the featured image for pages and categories.
You see examples of doing this with a custom walker, but the code is very complex to do for small changes. Surely there is a way to do it with a filter?
This is the code I came up with thanks to some help from a Wordpress StackOverflow answer that I can't find anymore (please comment with a link if you find it).
First you need to add the filter to the specific menu (you could add it to all menus if you want - just use the add_filter line by itself).
Then you need to build out the code to get the post or category ID from the $item object passed to the filter. It's not as easy as you'd expect, as $item doesn't contain the underlying post/category ID, just the menu item ID. So I use the URL's to do a reverse lookup of the IDs.
This won't work for tags used in a menu, or custom taxonomys. I only needed it for categories, so this is all I built.
And then you want to remove the filter that you applied at the beginning, so that the next menu processed doesn't use the same HTML as defined above in filter_menu_items().