I wasn't able to find any history of the sign through a quick google search, but while that remains a mystery, the sign alludes to another element of Kansas history: the evolution of the Jayhawk. The Jayhawk most often seen today looks like this:
The Jayhawk memorialized in the Jayhawk Tower sign is actually the 1923 version of the bird, seen here.
Since there aren't any birds that look quite like the Jayhawk, you probably surmised that the bird is fictional. Its origin is rooted in the historic struggles of Kansas settlers, and the term was probably coined in about 1848. The name combines two birds: the blue jay- a noisy, quarrelsome thing known to rob other nests, and the sparrrow hawk-- a stealthy hunter.
Durring the Civil War, the Jayhawk's ruffian image gave way to patriotic symbol. The Kansas Governor at that time raised a regiment called the Independent Mounted Kansas Jayhawks. By the war's end, Jayhawks were synoymous with the impassioned people who made Kansas a Free State. In 1886, the bird appeared in a cheer- the famous Rock Chalk Chant*** (and arguably the best cheer that has ever been created).
***All historical information and pictures are available on the KU website. If you want to more about the evolution of the bird from one generation to the next, click here.