This happens for content types that have the "Add a link into the menu for new content of this type" option enabled (menu settings):
- Vanilla Backdrop installation
- Create a page, with it's title set as "Page", and save -> a "Page" menu item is added to the main navigation menu of the site
- Edit the "Page" page, change it's title to "Renamed page", and save ->
Actual behavior
The page gets renamed to "Renamed page", but the respective menu item remains "Page"

If you edit the page, the breadcrumb shows the old name (expected, since breadcrumbs pick parts from the menu items), which is confusing:

Expected behavior
Things work the same way as the "Generate automatic URL alias" checkbox works (pathauto "persist" feature):
When the "Provide a menu link" option is enabled for the node, there is another "Update menu title when the content title changes" checkbox. When that second checkbox is ticked, the "Menu link title" field is disabled:

The menu item title is synced with the content title. Renaming the content also renames the menu item.
- If the user wants to "break the sync" between the menu item title and the content title, they need to untick that second checkbox.
Alternatives
- When the "Provide a menu link" option is enabled for a node, there is an info message shown, which explains that changing the content title will not automatically update the respective menu item in the navigation menu.
- When the "Provide a menu link" option is enabled for a node, there is a checkbox under the node title field, called "Also update the menu title":

Recent comments
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There is also Node Comments Search - this searches the node and all comments (default search is node and first page of comments) as part of the content search. Izzy's module above is a separate...
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I have created a new module comment_search
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