Gold Elite leading
When you get to the gold Elite on your way up through the main facility, try this. Jump over him whilst he's busy growling, and run like hell. Keep going past the cloaked Elites, and then down onto the beach. The gold should follow you, and you can expect to hear his sword whizzing past your ears every now and then.
You can potentially lead him all over the island. He'll even chase you underwater. If you can, take the opportunity of looking back at him as he does this; it's quite a scary sight to his outline looming out of the murky depths. Back on the beach, if you get to a Warthog then you won't have to worry about him being faster than you any more. Have fun making him leap out of the way as you gun the Warthog at him.
Instead of taking him outside, you can lead him down underground, making it a bit easier to keep ahead of him by being good at weaving around things.
Update, May 2014: I've realized that you can also get the Elite frozen outside, to launch him as a firework, in a similar way to Hunters (as described in Hunter fireworks). Expect an article plus movie footage in due course!
Update, January 2017: See Goldie fireworks.