I am having this issue as well - Mac 27" LED Cinema display.
I have read elsewhere that this is a power supply issue (the one inside the monitor) ...or sometimes a logic board issue. But several people who have switched out the power supply with a newer part design have had good results.
I really don't believe this has anything to do with the mouse, cord, plug, socket, software etc. I believe it's an issue with hardware inside the monitor itself.
You can buy a new power supply for $130 online, and you'll need a couple tools to get it open like suction cups to get the glass off and a Spudger to get it open, plus some star (Screw) drivers. While checking out the step-by-step tear down of the thunderbolt display on iFix-It I found they sell these tools and ordered them from the site. I'll let everyone know how it goes when I replace the PS. I hope that's it, I really don't want to have to replace the logic board too. 😟