The lexicon is the store of words in long-term memory. Any attempt at modelling lexical competence must take issues of string storage seriously. In the present contribution, we discuss a few desiderata that any biologically-inspired computational model of the mental lexicon has to meet, and detail a multi-task evaluation protocol for their assessment. The proposed protocol is applied to a novel computational architecture for lexical storage and acquisition, the """"Topological Temporal Hebbian SOMs"""" (T2HSOMs), which are grids of topologically organised memory nodes with dedicated sensitivity to time-bound sequences of letters. These maps can provide a rigorous and testable conceptual framework within which to provide a comprehensive, multi-task protocol for testing the performance of Hebbian self-organising memories, and a comprehensive picture of the complex dynamics between lexical processing and the acquisition of morphological structure.