Eighteen characters is all an owner gets when naming their equine athlete and perhaps establishing a moniker that evokes a legacy akin to that of Man o’ War or Secretariat.

In 2012, while 29 weeks pregnant, Hatch posted a photo on her former blog, “Mommy’s Little Sunshine.” Hatch posed in front of a chalkboard bearing baby name options for her now 13-year-old daughter, ...