This TARSK is one for the younger researchers. A hole count is a summary of market research data that simply counts how many times every break of every question has been picked by the respondents.
So, why is it a hole count and not a whole count? When I started working with data it was still sometimes being stored on punch cards, i.e. 80 columns of 12 rows, per card. When data was entered holes were punched in the cards (questionnaires would specify which card and which columns represented the answers to a particular question).
The hole count was a tabulation of all the holes on all the cards for all the respondents and was typically the starting point for quality control and for getting an overview.