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I also can't find much about her. Her name though may have been Lucia Gueth; , , and  indicate that Lucia Gueth (daughter of August Gueth) married a Lee Mida of Chicago in about the right time frame.
Lee W. Mida (her husband) was the Chicago City Champion in 1909 and she won an indoor golf tournament of some kind in 1913.  calls her one of the finest players in the Chicago district in 1914. The couple apparently played at Jackson Park.
I tried to verify the Time Magazine article. The WWGA has an amateur championship dating from 1901 and a list of champions here; the article mentions that Elaine Rosenthal won in 1915 and 1918, which matches the list, but the article has her winning again in 1924 where the list has 1925, and the golf courses mentioned don't match at all, so they may have been a different tournament. The list has no mention of Mrs. Lee Mida. This Time article has Rosenthal winning the tournament again in 1925 (in a field containing Mrs. Mida), and the course is different again, so Time Magazine was probably referring to a different tournament.
This eBay item is a newspaper portrait of a U.S. women's team which apparently had an unofficial match at Sunningdale in Great Britain in 1933; Mrs Mida is there along with several other notable golfers of the time (Glenna Collett, Maureen Orcutt, Virgina van Wie, Diana Fishwick, and others) many of which played on Curtis Cup teams in that era. About all I could find ;-) Carl Lindberg 06:43, 19 May 2007 (UTC)