Does My Baby Need Her Own Special Name?
–Like Mother, Like Daughter
It seems to me that your baby would already have her own name: Elliana. But let's look at the question of what constitutes an homage: What's the best way to honor your mom in her granddaughter's name? Generally, the standard is: Will the honoree (or if she is deceased, the people who loved and remember her) feel suitably honored by the name you choose?
In some cases, a shared initial, a modification or update (like Jackson to honor a Jack, or Caroline for Carol), or even a word or place associated with the person will do the trick. Practically speaking, though, the further you get from the original name, the less potent the homage. Maybe that's what you’re going for: a link, but not a direct one. In that case, Kate is great!
But if you'd like to really make clear that your baby is named for her grandmother—and assuming that your mom doesn't use the nickname Kate—you might stick with Kathryn. That's especially true with a middle name, where the bar is a little higher. There's less style pressure when you're not calling the child by this name regularly, so why not use the real deal?