Try this:
Settings...developer options...uncheck do not keep activities
Edit: If that doesn't work, try:
Settings > Application Manager > All > Messaging > Clear data
First one, I have no dev options turned on, but just double checked. No luck.
I tried the second one a while back, but no luck. Just tried again, and restarted the phone after deleting the icon, still no luck.
A google search yielded
go to application manager - all apps - "badgeprovider" - clear data - restart phone
as an option, but I am guessing that was a different version of android or a different phone, because I don't have "badgeprovider" in my list of apps, and couldn't find anything I thought was similar.
Motorola Droid Razr HD
Android version 4.1.2
Default text messaging app