If you're looking for a way to get rid of the extra tissue non-surgically, then I'm afraid you're out of luck. What you have is a common condition called keloids.
Successful removal of scar tissue from around an ear piercing requires surgical intervention. Call a board certified plastic surgeon; she will be able to assess your scar. Some people have reported moderate success with the application of vitamin E oil directly on the scar tissue. The oil may help reduce symptoms associated with scar tissue -- discoloration and itchiness.
Tea tree oil is one of the most common and cheapest remedies out there; try 2-3 drops once a day.
If it is hypertrophic scarring sea salt and Chamomile soaks should fix it.
You could ask your doctor about having it surgically removed, but in most cases the tissue will grow back and could possibly even increase in size.
Unfortunately the only way to get rid of scar tissue around an ear piercing is to have it removed by a doctor. A doctor can remove it similarly to how they remove a mole. Depending on the size of the scar tissue it may require a couple of stitches to heal properly.
Sometimes, scar tissue underneath tattoos or around piercings can swell up because of stress, exhaustion, sun exposure, or weather. You can alleviate swelling by resting and icing the affected area.
I'd really recommend you go see your doctor and ask about it if it's bothersome to you, he can recommend a surgeon that can cut it, or laser it. There's no home remedy for this, you need to see a doctor and let them decide the best method for you.
Without seeing a picture of the piercing, I am guessing that what you have is a keloid. While you cannot avoid keloid entirely, you can help avoid them by going to an actual piercer, and not someplace like Claire's.
