Maria McKendrick & Stephen A. Douglas(s) would share great-grandparents if they were second cousins.
Were Stephen & Martha (Gardiner) Arnold their common ancestors? The will of Stephen Arnold can be found here; granddaughter Martha Arnold, daughter of his deceased son Beriah, is named in the will. Other children of Stephen & Martha Arnold are named here.
From The Gardiners of Narragansett...:
Stephen & Martha (Gardiner) Arnold were the parents of Mary (Martha) Arnold. Mary married Benajah Douglas(s) and were the parents of Dr. Stephen A. Douglas(s). Dr. Stephen A. Douglas(s) married Sarah Fisk; they were the parents of Stephen A. Douglas(s), the politician, who was born April 23, 1813, in Brandon, Vermont.
A biography of Stephen A. Douglas(s) can be found here.
From Papers In Illinois History....
...the senator's talents were supposed to be mainly inherited from his grandmother, Martha (Arnold) Douglass, who is said to have been a woman of more than ordinary intellect and force of character.
The ancestors of Maria Arnold White McKendrick have not been identified, therefore I can't confirm or refute the assertion that she is the second cousin of Stephen A. Douglas(s).