Wendy Wear Stapleton Reveals Why Cousin Barbara Bush Chose Baby's Name: 'She Likes the Alliteration'

Barbara Bush's cousin Wendy Wear Stapleton is offering insight into her newborn baby's name.

Barbara, 39, welcomed daughter Cora Georgia, her first baby with actor-writer husband Craig Coyne, on Monday.

"She likes the alliteration, the Cora Coyne, so I know she was tossing around C names and that was kind of in the lead in late August, so I'm glad she did that," Wear Stapleton exclusively told PEOPLE at the third-annual George H.W. Bush Points of Light Awards at Casa Cipriani in New York City on Tuesday.

Wear Stapleton also shared why her cousin chose her daughter's middle name for her father, former President George W. Bush, and her grandfather, former President George H.W. Bush: "Then she's got the Georgia which is for her dad and her grandfather."

She also gushed about how she feels about where Cora Georgia was born and the anticipation of their first encounter: "I think it's so cool that she was born in Kennebunkport [in Maine] because we're just kind of a tribe, and I can't wait to meet her."

The Bush family has consistently spent their summers in Kennebunkport, as their compound on Walker's Point is also located in the northeast coastal town.

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Nicholas Hunt/Getty Barbara Bush

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During the event which honored well-known change-makers making an impact in their communities and the world, Barbara's uncle, Neil Bush, revealed that Cora Georgia was born premature, which kept the new mom from attending the event.

"She was six weeks early, otherwise she would have been here full of baby," he said.

Neil's daughter and cousin to Barbara, Lauren Bush Lauren described the euphoric mood of the new parents to PEOPLE, "They seem very smitten and happy."

What kind of mother does mom-of-three Bush Lauren believe Barbara will be? "Wonderful, loving — a very cool mama!"