I would start by checking to make sure the by the mouse command is still bound to the correct mouse event. Under SlickEdit Preferences > Keyboard and Mouse > Key Bindings, you can either search by command (mou-select-word) or by key sequence (use the mouse button to select mouse event LDoubleClick). mou-select-word should be bound to LDoubleClick for the default mode. If it is not, then you'll need to re-bind the command. The only other setting that might cause problems is if the Word chars setting for whatever language mode you are in is corrupted or invalid. For example, the default Word Chars setting for Java mode (SlickEdit Preferences > Languages > Application Languages > Java > General) the Word Chars setting is A-Za-z0-9_$.
The nuclear option would be delete your existing config file and restart with fresh default config to see if the issue resolves. Configuration Directory would be listed in Help > About SlickEdit Core > Program Information tab. You may want to back up your configuration directory rather than delete, or try Export/Import feature in SlickEdit Preferences, but the bad setting may be there on import. Depends on how many modifications you have made to the default configuration.