To confirm you mean the Display Widgets Options (like the hide/show options) aren’t saving rather than widgets generally aren’t saving. If it’s widgets generally not saving (like adding some text to a text widget doesn’t work) it’s unlikely to be the Display Widgets plugin: there’s lots of things that can mess up the widgets area. For example a few months back I upgraded to a new server, everything seemed to be working fine other than on multisite WordPress installations I couldn’t open widget areas at all! Tracked it down to forgetting to increase the memory limit on the new server, WordPress was running out of memory on the multisite installations, so I couldn’t work with widgets at all (simple increase in memory fixed the problem).
Anyway, assuming it’s the Display Widgets options not saving…
Nothing obvious comes to mind.
I’m afraid it’s general advice to track down problems that could suggest a conflict with the theme or another plugin. WordPress users tend to update the WordPress version and themes/plugins at the same time, so could be an updated theme/plugin causing a conflict rather than the WordPress update per se.
Try one of the default themes – this rules out a theme conflict if you experience the same issue while using say the TwentySixteen it’s not a theme specific issue.
Disable plugins one by one – rules out a plugin conflict.
Try a different browser (Firefox, Chrome) – rules out a browser specific issue.
If you have access to your sites log files, try accessing the Widgets area then check your logs for errors that might suggest an issue to troubleshoot (how I found the memory issue mentioned above).
Good luck.
David