In general cytokines are named after their presumed function, or cell secreting it, or their target of action.
For instance the term “chemokine” refers to a specific class of cytokines that mediates chemo attraction (chemotaxis) between cells. The word “interleukins” has been used to refer to the cytokines acting as mediators between different leucocytes/lymphocytes.
The term “lymphokines”, has been used to refer to the cytokines released by lumphocytes.