Deliberate mistake: In the very first episode, the kids are fighting and arguing all throughout the house. Claire then asks Cliff, "Why did we have 4 kids?", referring to Denise, Theo, Rudy, and Vanessa. In all of the following episodes, they actually have 5 kids. I guess they decided to add Sandra (the oldest Huxtable) later down the road.
Continuity mistake: The father was known as Clifford in the beginning but the name was changed to Heathcliff (you can see this on the nameplate for his office).
Continuity mistake: After Theo gives up an extra $100 for clothes, he is holding $300 left in his hand. However when the camera switches back to Cliff, who asks how much is left, it switches back to Theo, who is only holding $200 left.
Chosen answer: That is the song "Open the Door Richard."