How can I dodge 200 lightning bolts in a row?

There is, in fact, an easier way to do this! There is a specific area you can go to in the Thunder Plains, and cause the lightning bolts to hit the same spot, over and over. This makes it much more consistent to dodge, since the timing is no longer random and much easier to predict.

The area is just outside of Guadosalam. If you've activated three of the Cactuar statues, the little ghost is running around just south of it. It looks like a little hollow, right by a tower. The trick is to walk just over the dip in the ground, and that's when lightning will strike. Press X as you move into the dip, and you will dodge it. The timing is pretty generous, I've found; you can move into the hollow, and press X before the flash, and you'll still dodge it. Then it's just run to the tower, loop around, and go at it again.

Doing it this way gives you one lightning bolt about every three or four seconds, which means it'll take around ten to fifteen minutes to amass the necessary 200. Note that you won't have any indicator that lets you know when you hit 200; you have to go to the Travel Agency and read the book to see how many you dodged in a row. But don't go there until you miss, because leaving the area will reset your count.

To see this in action:

Frank's answer is the correct one. But I wanted to add some points that helped me:

1. As I said in this comment you'll wanna do this after completing your captures in the Cavern of the Stolen Fayth. That way you will have fought enough Ghosts to obtain No Encounter armor for one of your party members. The armor should be equipped before attempting this side-quest as lightning can strike immediately after a battle.
2. There is some advice out there recommending dodging lightning in the crevice with the Cactuar Stone. This is where I accomplished this side-quest in the American Version of Final Fantasy X. The International version tweaks the timing though, making it nearly impossible to dodge lightning in said crevice. So stick to the crater location in Frank's answer.
3. The distance you run from the crater determines how long it will take you to complete the side-quest so don't run all the way to the tower each time You only need to go about half way there.
4. Trying to reach 200 with out taking breaks can be grueling and will wear down your concentration. I reached 200 the first time I decided to take breaks every 20 dodges (without taking breaks I was averaging about 60.) You can take a break by pausing or just standing under the tower. Do not take a break by just standing half way to the tower. Lightning can still strike there randomly if it hasn't been forced to strike by running to the crater recently.
5. Lastly, you do not get the Lightning Dancer Trophy until you actually obtain the Venus Sigil in the Thunder Plains Travel Agency, so you must keep count for yourself, don't expect an achievement notification to let you know you're done.

http://www.psnprofiles.com lists the Lightning Dancer Trophy as more rare than even the Chocobo Rider Trophy, which seems astounding to me. (Of course they also list it as more rare than the Weapon Master Trophy, so maybe take those numbers with a grain of salt.) Either way this can be a difficult quest and I hope my tips have helped you. Good luck!