What is this part that came with my eggbeater pedals?
Without a picture it sounds like the grease port tool

This picture was found in this article on rebuilding eggbeater pedals.

The Crank Brothers Eggbeater pedals, however, require periodic lubrication. A maintenance interval of once per season seems to be the general consensus.
As the Eggbeaters come with a grease port attachment, I purchased a grease gun

Evidently Eggbeater supplies the fitting but someone there does not recommend using it.

It seems like a painless, 5-minute job: remove dust cap; install grease port attachment; squirt in new grease, wipe off the old grease (pushed out from the seals on the inboard side); reinstall dust cap; and you are done.
Call me anal, but I decided to double-check with the manufacturer — and I am glad I did:

Hello,
Thank you for your e-mail.
...we would suggest not using the grease port. We would rather you take the spindle out and  grease it like that. Just unscrew the endcap and take the nut  off -  then pull the spindle out and re-grease.
Hopefully this helps,
Thanks
Anka Martin
anka@crankbrothers.com
Customer Service
www.crankbrothers.com


Here is another article with slightly better pictures.