It maybe that your system preferences plist has become corrupted.
The apple.com.systempreferences.plist is in in the following path.
Macintosh HD/Users/Home/Library/Preferences
The Users Home Library is hidden by default, to view this Library click on Go in the Finder menubar, hold down the alt/ option key and you will see Library will be revealed under Home, click on Library and a new Finder window will open
navigate down to Preferences then drag com.apple.systepreferences.plist to the Trash. Don't delete just yet.
Restart your Mac open System Preferences select the desktop wallpaper you want to use. Restart your Mac, has the new wallpaper been remembered. If it has and you are happy then open System Preferences again as you will need to
go through each preference field and customise as to your liking. Once you have done that and things are back to normal you can delete the older system preferences plist that is in the Trash.
If you are determined to throw your laptop out you can send it to me.