Okay, that’s cute.
As a horror show, Doctor Who tends not to welcome new births with glad tidings and cake. Amy was hit by a bunch of expectant mothers’ worst fears. Gwen didn’t have much luck on her wedding day in Torchwood either. Births on screen tend to be monsters hatching. About the best we’ve seen is
One notable exception is Luke (and later Sky) from The Sarah Jane Adventures, where parenthood is something heroic to be celebrated - and of course they’re teens pretty much as soon as they arrive. (Babies turning into young adults who can take on the role full-time at great speed is so practical that it’s all over fantasy and SF TV and you can find it outside of genres where it fits too!)
The Moffat years have been notably kid-friendly, from wee Amelia Pond to Stormageddon to Artie and Angie. If you want a distinctive Eleventh Doctor feel for a particular adventure, consider having a couple of kids among the characters who might act as temporary companions...
Birth itself could be a source of adventures too.
The predicted birth of a child destined to change the world? That might be a bit on-the-nose, but if the players wouldn’t mind a “guard mission”, the mother of an unborn Messiah or Antichrist (opinions vary) pursued by various forces would certainly make for an interesting one.
The mysterious deaths of everyone in the city born on a particular day, in the order they were born in. And one of the PCs was born late that night...