Oh well, now I just read a post where it was stated that these gay Republicans believed Lincoln was gay and named themselves after him without using his name...using his association in history with a log cabin, thus calling themselves Log Cabin Republicans. Maybe that's it and I dreamed that it had to do with their meeting first in a log cabin. I'm really not sure, just thought I recalled something...
To be honest it's a guess on my part. It's not something I have investigated.
Thanks.
Someone posted a link to wikipedia:
The name of the organization is a reference to the first Republican President of the United States, Abraham Lincoln, who was born in a log cabin. The group initially proposed to name themselves Lincoln Club, but found that name was already in use by another California Republican organization. Thus, an alternate name was chosen that still invoked the memory of President Lincoln: Log Cabin Republicans. Historical accounts by biographers that Lincoln may have been gay surfaced long after the organization was formed, and did not contribute to the choice of the name.
http://en.wikipedia.org/wiki/Log_Cabin_Republicans
Not sure if that is accurate either, but I have at least satisfied my linguistic curiosity. :-)