Can I promote someone else to be a group admin on Facebook?

I'm not positive I've figured out the solution to your dilemma, but I have a pretty darn good idea how you can proceed. The basic procedure is that you need to have the other person join the group then you make them an admin and resign from the group yourself.

Here's how I added my friend Dan to a group I run on Facebook, as a demonstration.

First step is to figure out in the new "improved" how to get to Facebook groups. The easy way is to click on the "groups" icon on the lower left corner of any Facebook page:

Now you're in the "Groups" area, click on "My Groups" on the top left:

Now choose "Groups You Admin" from the different choices you have:

Finally, you should see your group (or groups, if you have more than one):

Okay, now we can do something! Click on "Edit Group" and you'll see a page that has a variety of different information about your group:

At this point you want to click on "Invite" (though if you want to make someone an "officer" of the group along the way, click on "Officers", but that's another story):

Notice that you'll see a list of every member of the group, and adjacent to each "Remove" and "Make Admin".

Find the person you want to promote to administrator, then click "Make Admin".

You'll get a verification window:

Ready to do it? Click on "Make Admin" and they're now an administrator for the group.

If you look at the group info page you'll now see the new admin (in this case, my friend Dan Gray) show on the group info page:

If you decide you don't want to keep them as an admin, just click on the "Remove Admin" link:

And, as usual, you get a verification window:

If you want to proceed, make the new person an admin, then simply remove yourself from the group entirely. At that point, they should be the only admin and you should be free (though I have to admit I have never tried that very last step. Perhaps someone else who has more experience with Facebook groups can comment on whether this does in fact work?)