Mark groups as favorite

That would be a very helpful feature. Not sure how feasible it is to implement though.

Just checking: on GA creation, there is a real-time filter box for the groups you are seeing, just type a few characters in the group's name.

I do not mean to invalidate your suggestion, just wanted to make sure you knew this workaround (you probably did, I guess).

Cheers!

ESGST has this feature if you use it, or want to use it:
(2.43) - Stickied Giveaway Groups (this url will work to take you to the exact setting if you have ESGST installed)
What does it do?
Adds a button ( if the group is stickied and if it is not) next to each group in the new giveaway/groups pages that allows you to sticky the group so that it appears at the top of the group list for quick use.

While it seems the extension version has compatibility issues now, you can still download the userscript version.