CacheName is the name of an optional argument, whose value is either a mutable hash table or null. After each stage of lifting, the methods versalDeformation and localHilbertScheme will store the solution to the deformation equation in CacheName#VersalDeformationResults. If the value of CacheName is null as is the default, then the solution is stored in the hash table described in the documentation of versalDeformation and localHilbertScheme.
The source of this document is in VersalDeformations.m2:1255:0.