What is the name of Big Bird's teddy bear, the one he sleeps with?

RADAR.

http://www.members.tripod.com/~Ashey2/bigbird.html

Radar is Big Bird's teddy bear on Sesame Street. He's a brown stuffed bear with a red ribbon around his neck, who was given to Big Bird as a special gift from Mr. Hooper. Much like Ernie's Rubber Duckie, Radar is Big Bird's most treasured possession, and Big Bird often confides in the bear.

(NOTE: It is also rumored that Radar was given to Big Bird by Gary Burghoff (who played Radar on M*A*S*H) when he guest starred on the show -- UNSUBSTANTIATED).

http://muppet.wikia.com/wiki/Radar

https://en.wikipedia.org/wiki/Big_bird