What does it mean to say that a strong base is only slightly soluble?
Solubility is different from dissolution, which is itself different from dissociation.

Solubility tells you the extent to which a substance dissolves. Dissolution is the kinetic process of solvation, when solvent molecules interact with solute molecules or ions. Dissociation is, from IUPAC, 


  
  The separation of a molecular entity into two or more molecular entities (or any similar separation within a polyatomic molecular entity). Examples include unimolecular heterolysis and homolysis, and the separation of the constituents of an ion pair into free ions.
  The separation of the constituents of any aggregate of molecular entities.
  


Dissociation is the mode of solvation for ionic compounds.



I think your confusion stems from associating solubility and dissolution improperly. ChemGuide directly answers your question here:


  Some strong bases like calcium hydroxide aren't very soluble in water. That doesn't matter - what does dissolve is still 100% ionised into calcium ions and hydroxide ions. Calcium hydroxide still counts as a strong base because of that 100% ionisation.