I found an Elizabeth Peters listed as a pupil in the Secretary's Book in 1862 for the Sargent Chapel and the Pleasant Grove Sunday School class. There were no Brannons listed.
I don't know if you have this or not, but there is an Elizabeth Peters on the 1850 Coles County census in Independence Precinct, when would be in the same area as what is now Douglas County.(Douglas County was formed from Coles County in 1859.)The 1850 listing reads: Stephen Redden,30,KY; Vashti Redden,26,IN; Elizabeth Peters,5,IL.
Elizabeth Peters age 14 is still with Stephen and Vashti Redden in Camargo,Douglas County in 1860.
Stephen Redden married Vashti Winkler in Coles County 3/23/1840.I would guess that Elizabeth Peters is some relation to one of them but I didn't find an obvious link online.The library might have more information if you'd like me to look. My email is firstname.lastname@example.org.