Joan insists, without allowing for any argument, that Morse choose her daughter's name. If he's going to raise her, then there's going to be something she can point to and say 'that's from my dad' even if it's not genetics. If she's not going to share his hair or freckles or eyes, at least she'll have the name he chose, because he wanted her. And that's how she ends up being named Verity. And I love that.
Joan grows to love that child. Until she's nearly two, she feels horribly guilty about how little she feels connected to her, but once Verity starts talking, that changes everything. Of course, Joan still works, and honestly, it makes sense that Verity is a bit of a Daddy's girl. Morse always has been more openly affectionate, had an easier connection. But Joan does develop a good, close relationship with her, especially as she gets older.
Verity Morse likes her dad's silly operas. She likes poetry and reading. And she likes football and, strangely, boxing. When she's little, she wants to grow up to be a racecar driver. She actually ends up pursuing a degree in mathematics.