In I Love Lucy Series 6, episode  18, “Lucy Gets Chummy With The Neighbors” we see Lucy’s handy flip up telephone index. In the pre-digital era  this type of mechanical index could be found on office desks and in the home next to the telephone. A closer look at the detail  makes it easier to identify thanks to a Jamie who has indeed located one just like it on eBay. It is a  List Finder / Telephone Number Organiser marked “Made in USA” on the back but no brand name is seen.  Says Jamie ” I know these were initially made by a company called “Bates” and called the “List Finder” back in the mid 30s and continued all the way into the 60s. There was also another company that made a different style called the Zephyr Autodex.
The particular one from this episode doesn’t appear to be made by either, but I did find one of the episodes style on ebay. The only markings on it are “made in USA”. , but it definitely is the one!”

More details and pictures on eBay https://www.ebay.com/itm/401124154834

Thanks again to Jamie for a great find!