Horses are extremely sensitive and though it might not seem like it your horse can feel even the slightest movement in your body even if you don't know you are doing it. One of the things I have done with my horse is making a figure 8 using the whole arena at a canter. Then in the middle bring your horse back to a trot and then turn the opposite way and pick up that canter. The way your body is going to move without realizing is something your horse will pick up on. Eventually your horse should do this automatically.
However, all of these are awesome tips that work well for horses that CAN lead change. My horse has the improper conformation for lead changes. He is simply unable to flying lead change no matter how hard he tries
One way is ride your horse diagonal on the right lead and when you get on the other side of the arena go left and they should naturally change leads. Then start using the cues at that time and it shouldn't click.
Im assuming this horse already knows changes of lead through the walk? That comes first. For flying changed once you have a well balanced & collected canter, just ask for the new lead by half halting, and giving the new aid. Very simple, but some horses take longer than others to figure it out. You have to know how hard to push, and when to back off if you're going to be teaching a change. It takes a very precise timing.