What is the name of the small plastic disc inserted into the center of a 45 record to allow it to fit onto the spindle in the center of a turntable?

It was known as the spider.

It is imaginatively called a 45 R.P.M. adaptor in several hi-fi manuals. Apparantly sometimes called ads as in "adaptors".

The former RCA Corporation introduced a snap-in plastic insert known as a spider to make 45 rpm records compatible with the smaller spindle size of a 33 rpm LP record player. Commissioned by RCA president David Sarnoff and invented by Thomas Hutchison, spiders were prevalent in the 1960s and sold tens of millions per year.

The Hutchison adapter included small bumps called "drive pins," which locked the adapters together while revolving, thus preventing the stacked records from slipping against each other. Several manufacturers made "spider" adapters in slightly varying shapes and many different colors, though yellow and red were most frequently used.

https://en.wikipedia.org/wiki/45_rpm_adapter

When I worked in a record shop in the 1970's, these were called spiders.

Being a fan of good old vinyl (what exactly ARE these new compact disc thingies?) I have known them as 'spiders' though my mother has always referred to them as 'middles.' They vary in quality - some are either too large or too small for the holes in the centres of the records, while others have a central hole that is too large to fit snugly on the turntable. In the UK, when seven inch single records were the dominant format, the majority of them were manufactured with only a small hole, thus eliminating the need for spiders. These little devices were useful for imported American singles, or ones where the holes had purposely been enlarged for use in Jukeboxes.

I was a DJ during the fifties at a five thousand watt AM station and the jocks all refered to them as "ads"; short for adaptor.