When Frank Hudock, 35, a restaurant manager in the Chicago area, first told his wife, Jennifer Hudock, 30, an executive assistant, that his grandparents wanted them to name their son Frank, per generations of family tradition, her response was, “That’s never going to happen.”

They had just decided on a name they both agreed on — Max — after arguing about dozens of others, and that had been a big relief.

But then the grandparents threw in a sweetener: an offer of $10,000 in exchange for choosing Frank.

Ms. Hudock’s company didn’t have a maternity leave policy, and she wasn’t going to get paid for several months. She agreed to think it over.

This may seem like an unusual proposal, but several professional baby-name watchers report seeing others along those lines. Maryanna Korwitts, a naming consultant also based near Chicago, said one client’s grandfather offered a family business if a baby could be named for him. Another’s in-laws, she said, suggested a dream wedding the mother never had and could not afford on her own — the price being the right to name the firstborn.