“Brother against brother” has been used so much it’s a cliché, but that’s because it accurately describes the experiences of many Civil War-era families.
Perhaps the most notable family was Abraham and Mary Todd Lincoln’s family. The Lincolns and Todds both had roots in slaveholding Kentucky, and Mary’s brothers all fought for the Confederacy. David Todd treated Union prisoners brutally, which deeply embarrassed the Lincolns. Samuel Todd was killed at Shiloh. George Todd was implicated after the for abusing Union prisoners as a CSA doctor. Alexander Todd was in Kentucky’s Orphan Brigade and was killed by friendly fire in Louisiana. And Mary’s brother-in-law was killed at the Battle of Chickamauga. The First Lady’s rebel family was one reason Washington Society treated her badly. Historian Stephen Berry has a very good and easy to read book about this called “House of Abraham: Lincoln and the Todds, a Family Divided by War.”
Historian Amy Murrell Taylor wrote what’s likely the most notable book on divided Civil War families in “The Divided Family in the Civil War.” She examines the emotional, political, and psychological implications of fighting a war against you own family. She argues that divided loyalties among kin help make the war more brutal and traumatic for many people. Historian Victoria Bynum also shows that it could cause long-lasting conflicts between family members in “The Long Shadow of the Civil War.”
There are a few notable examples that come to mind:
Families from border states (Maryland, Kentucky, Virginia, West Virginia, Kentucky, and Missouri) were much more likely to have divided loyalties, so local histories from those states are a good source of you’re looking for examples.