The Confederate Constitution provided that "No person except a natural-born citizen of the Confederate States, or a citizen thereof at the time of the adoption of this Constitution, *or a citizen thereof born in the United States prior to the 20th of December, 1860,* shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained the age of thirty-five years, and been fourteen years a resident within the limits of the Confederate States, as they may exist at the time of his election." [my emphasis-DT]
http://avalon.law.yale.edu/19th_century/csa_csa.asp
Theodore Roosevelt was born in the USA (to be more precise in New York City) prior to December 20, 1860 (to be more precise, on October 27, 1858). So he would have been eligible for the Confederate presidency, had the South won the ACW. A possible POD: Theodore Roosevelt, Senior, dies much earlier than in OTL; friends think that it may be due in part to his sorrow at the Confederate victory. His widow, Martha Bulloch Roosevelt ("A true southern belle"
https://en.wikipedia.org/wiki/Martha_Bulloch_Roosevelt) who never made much of a secret of her Confederate sympathies, decides to return to Georgia, taking her young son Theodore with her. I doubt that under those circumstances, TR's northern birth would be much of a political handicap in Confederate politics, even if CSA-USA relations are bad. As he could point out, he was born of a southern mother, lived in the South since he was a boy, grew up on a plantation, etc. It's even possible that schoolmate taunts about his being a "Yankee," a "black Republican" etc. will make him *particularly* racist and anti-Northern...