Betsy Martin was born in Athens County, Ohio, about 1825 or 1826, I believe. That is, I have inferred that she was the eldest of the three girls "under 5 years of age" listed in the household of John and Sarah Martin in the 1830 census. (See my discussion of Louisa Martin for an explanation of the reasoning behind that conclusion.) If Betsy had had her fifth birthday prior to the enumeration date for that census (June 1, 1830), she would have been listed in the "5 to 10" age range. On the other hand, various censuses suggest that her younger sister Lucinda was born no later than 1828, thus it seems unlikely (though not impossible) that Betsy could have been born much after the end of 1826.
Betsy would have been of marrying age during the 1840s, by which time the family had moved to Coles County, Illinois. The on-line Illinois marriage records do not show any marriages involving a "Betsy Martin" around that time, but they do show two brides named Elizabeth Martin who were married in Coles County in the 1840s. One married Ignatius Levett on 22 December 1845. This bride does not seem to be John and Sarah's "Betsy." The 1850 census shows Ignatius and Elizabeth Levett (or Leavett) living in Effingham County, Illinois. The Elizabeth in that record is 21 years old (a bit young for our Betsy) and, more importantly, was born in Iowa.
The other record of interest in the on-line index shows that Elizabeth L. Martin married Alfred Wright on 1 October 1842. However, in the handwritten register of Coles County marriages, the groom's first name is shown as Alford (and the actual marriage date is 3 October). Regardless, though, of whether the man's name was Alfred or Alford Wright, I have been unable to identify him or his wife in any census records, so I cannot confirm whether her age and place of birth are the same as Betsy's. I will welcome any additional information anybody can provide about Betsy and/or about Alford and Elizabeth Wright. My contact email address is shown in the image below.