Is it JUST a name?

We asked my MIL to name our next daughter and she chose the name "Josephine". I personally think it's a beautiful name BUUUUT my husband was quick to respond with and I quote "What the fck Mum? No! We're not naming our child that!" Why, you ask? Oh, because it's his ex-fiancées name 😆 to which my MIL replied "Stop being so melodramatic. It's just a name.." 
Is it really tho? How would feel giving your child the same name as your SO's ex?