I had to answer this post because my name is Hannah (well, it is Hannah-Sue, but everyone calles me Hannah). First, I think it is a name that might not date so much as other popular ones now because it has been popular for quite a while. The negative side of this is of course that having been popular for a while, everyone knows quite a few Hannahs and one ends up being known as Hannah A., Hannah B., or in my case Tall Hannah.
I'm not great at suggestions, but I'll have a bash!
Hannah Lisette
Hannah Rosaline
Hannah Charlotte
Hannah Iris
Hannah Odette
Hannah Esme
Hannah Estelle
Hannah Lilac
Hannah Celeste
Hannah Maeve
Hannah Evelyn
Hannah Beatrice
Hannah Delphine
Hannah Eugenie
Hannah Madeleine
The Hannahs that I know have the middle names Louise and Kate.