What is the proper name for the 45 RPM record insert that lets the record be
played on an auto-drop 33 RPM spindle record player?

I'm giving my age away, but I used to have hundreds of 45 rpm records,
and an equal number of those little gizmos that fit inside the hole of
the 45! When I was growing up in the southern U.S. in the 1950s, my
friends and I used to call these devices "spiders." When I was in
college in the 1960s, we called them "Triskelions," after an episode
of "Star Trek" which featured a design that resembled the little
plastic thingies.

In case you're curious, this is a Triskelion, as shown in "Star Trek":

Star Trek Dimension: The Gamesters of Triskelion
http://www.stdimension.de/Guide/sttos/tos046.jpg

And, for the very young, here is an image of the 45 rpm hole-filler in question:

Turntable Basics
http://www.turntablebasics.com/images/45adapter.jpg

On the Web you'll see these doohickeys called by many different names:
"45 inserts," "45 adapters," "spacer disks,"  "45 single adapters,"
"45 rpm record inserts," and on and on.

This comes from a site called 45s.com, a great place to buy vinyl 45s:

"Q.    What is the little plastic insert called that goes in the
middle of a 45-rpm record so it will play on a turntable?
 
 A.    It's called a 45-rpm adapter.  (This is the question we are
asked more than any other.)"

If 45s.com says so, I think that's definitive. My final answer: 45-rpm adapter.

Incidentally, you can still buy the little yellow plastic whatsits.
They cost $4.75 for a set of 25:

Baby Boomer Headquarters
http://bbhq.com/45adp1.php

Or, for $299.95, you can get a 45 rpm adapter made of 14 karat gold:

The Hackensack Record King: 45 RPM Adapter Pendant 
http://members.aol.com/records123/gold.html

It should be noted that there was a different type of device which
went on the record changer's spindle rather than in the hole of the
record. This was called a "45 rpm automatic spindle adaptor." Here's
an image that shows a collection of these. And, in the lower right
corner, of the image... the ubiquitous 45 rpm adapters, here called
"plastic adapters."

Garage-a-Records
http://www.garage-a-records.com/new/jpgs/adapt.jpg

Google search strategy:

Google Web Search: "45 rpm" +  "insert OR adapter OR adaptor"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=45+rpm+%22insert+OR+adapter+OR+adaptor

Thanks for a fun question! If anything is unclear, or if a link
doesn't function, please request clarification; I'll be glad to offer
further assistance before you rate my answer.

Best regards,
pinkfreud

Not exactly what I was hoping to find but a very well researched and
complete answer. I guess my mind is not as sharp as it used to be.

Thank you very much for the five-star rating and the tip!

On the matter of minds being less sharp than they used to be (and I'm
certainly aware of that problem!), my late father-in-law used to say
"My mind may not be as sharp as it once was, but it's well-padded and
more comfortable." ;-)

~pinkfreud