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I did a search on ILS in Google earlier.
The top result was for a website called ILSMart.com where the ILS stands for Inventory Locator Service. It is a supply chain company for the aviation and allied businesses.
Inventory Locator Service is not a bad name for what we call the ILS?
The first library-related entry was the fourth, which was the Wikipedia entry for ILS (Integrated Library System).