Setting DuckDuckGo as the default search engine works fine for me and sticks through a restart of Opera and a restart of my computer.
Do a test. Make a standalone installation to a folder named "Opera Test" on your desktop and test in that Opera. Does the setting stick there? If so, it's probably just something messed up with your normal Opera profile.
Do you use Opera Sync and have it set to sync settings? If so, It could be an issue/bug with sync messing up the setting. You could log out of sync for a while and see if the problem goes away. If so, then you know it's sync causing the problem.
Either way, you can try this if you want. Goto the URL opera://about and take note of the "profile" path. Then, close Opera. Then, load the "Preferences" file in the profile folder with JSONedit in tree mode. Right-click on the default_search_provider and choose "delete selected node". Repeat for default_search_provider_data. Then, save changes. That will delete your changes of search engine settings and hopefully fix the problem.
Besides that, I'd uninstall Opera (don't choose to delete your data when uninstalling) and reinstall Opera. That should repair any program file issues and the default search provider file. It might help.
