If someone is "as stiff as a board" it means that they are inflexible, and it usually means that they are referring to their back being difficult to move. Lumbar is an adjective that references the lower part of someone's back. Lumber is wood that has been processed into beams. The joke is playing on the word "lumbar" because it sounds like "lumber" and the person is "as stiff as a board". 
If you ' re stiff as a board it could be the lumbar .