Hi J. ~
My daughter loved her binky. She had one in every room, and couldn't sleep without it. Once she turned 2, our doctor said that it (or thumb sucking) could affect her front teeth and possibly cause an overbite in the front.
We had a hard time getting it away from her. Every time I thought I had taken them all away. She would find another one.
Anyway, we ended up leaving it for the 'binky fairy' who came while she was sleeping. The binky fairy takes all the big kid's binkies and delivers them to all the brand new babies. She then leaves you a present in return. We gave her a little gift bag filled with a big girl toothbrush, bath bubbles, bracelet, etc. She loved it, and we never had another problem with the binky.
Good luck!