According to this Wikipedia article:

http://en.wikipedia.org/wiki/Whittaker_model

Whittaker models are called that way because Jacquet pointed out that Whittaker functions appear naturally in the Whittaker model of a representation of $SL_{2}(R)$. (Assuming that the representation has a Whittaker model.) Is there a similar reason for naming Bessel models that way? That is, Is there a simple example where the realization of a representation with a Bessel model involves Bessel functions?