Hal's first name was from his father Henry; his middle name came from Bush Hager's grandfather Harold Welch, Laura Bush's father.
Hal joins big sisters Mila, 6, and Poppy, 3, and Bush Hager told her "Today" co-workers that the girls helped come up with what they were going to call their new brother.
She joked that her father was upset that they didn't choose George but liked the name they chose, USA Today reported.
"We weren't exactly sure about the name but then his big sisters called him 'Hal pal,' and we thought 'This is just perfect,'" Bush Hager said.
Bush Hager announced her pregnancy in April after she was named the co-host of "Today's" fourth hour, sitting at the desk with Hoda Kotb, the morning show reported.
Kotb went on maternity leave shortly after the new team's debut after she adopted a second child, People magazine reported.