Also check the settings in System Prefs->Mouse. You can adjust what happens when you "right-click", even turn it off. You can always use 'control'- click. Might also check that the battery contacts, in the mouse, are clean.
Finally, make sure you haven't gradually changed how you hold the mouse. Over time, you may have started using the "right-click" area of the mouse without realizing it.
Remember, these mice have mechanical switches. Switches fail, corrode, etc. They don't usually last as long as electronics.
BTW, if you're thinking of replacing the mouse, I highly recommend the Magic Trackpad. I rarely use my mouse now! But my rather large hand made the Magic Mouse useless. Your milage may differ!