Stephens got to a point where he "snapped and turned into a whole different person," his mother said.
"That is not my son," Maggie Green said.
Green and Stephens' brother, Ricardo Green, spoke with CNN on Wednesday.
Stephens killed himself in Pennsylvania on Tuesday as he was being chased by state troopers. He was wanted in the death of Robert Godwin, 74, in Cleveland. Stephens posted video on Facebook of Godwin's death.
After his suicide, Pennsylvania State Police recovered a .45-caliber Glock handgun, believed to be the murder weapon, in Stephens' car, and, according to a law enforcement source, a .380-caliber pistol at the home of his ex-girlfriend, Joy Lane.
Maggie Green, who expressed her condolences to the Godwin family, said Wednesday that she believes Stephens' breakup with Lane triggered the killing.
Stephens, who had attended an Easter Sunday service before the shooting, demanded Godwin repeat the name of his former girlfriend.
"She's the reason why this is about to happen to you," Stephens said on the video.
"I guess he just gave up," Green said of her son. "I guess he wanted to be caught."
After the Godwin shooting, Green said she called her son, whom she described as calm. She mentioned his Facebook posts and Stephens claimed to have killed 14 people. He was a "big exaggerator," she said.
Joy Lane, Stephens' former girlfriend, told CNN affiliate WJW
she was devastated the victim said her name before he was shot.
"I feel bad. ... The last thing that he would have said was my name and (he) didn't know me or why he was saying it. And that's been difficult," Lane said.
Lane said she and Stephens dated for a while, and had discussed getting engaged, but they broke up and she urged him to seek help for gambling issues.
Maggie Green said of Lane: "It's not her fault. It's not my fault."
Ricardo Green said his brother hated Cleveland and wanted to be taken seriously.
The killer often gambled in Erie, Pennsylvania, and he slept in his car when he went there. Green said on a December 2015 casino trip they took together, Stephens emptied his banking account, waited overnight until more money became available and went back inside, only to lose more cash.
Green said he gave his brother money for gas and food when the latter went bust gambling in February.
After the Godwin killing, a nationwide manhunt ensued, ending in Stephens' death not far from where he was spotted at a McDonald's
"I'm sorry it ended the way it did," said Maggie Green. "I know I couldn't sleep. I was up night and day until I see how my son was gonna leave this world."