From Wikipedia: The Log Cabin Republicans (LCR) is a federated gay and lesbian political organization in the United States with state chapters and a national office in Washington, D.C. The group consists of supporters of the Republican Party who advocate for gay and lesbian rights and has long been formally committed to doing so.
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. Further, by coincidence, Lincoln is the first President to have his sexuality called into question because of his relationship with Joshua Speed. See Sexuality of Abraham Lincoln for further information.