It would probably help to know a little more about the character's personality before we name her.
I mean regardless she'll get a new cutesy nickname courtesy of her caretakers, but still.

Honestly, don't have a whole lot established for her yet. I've imagined her to be a bit of a shy person though.

So whatever the name is it needs to be able to be either said cutely or be shortened for a more childlike variant for maximum blush-inducement.
Like say "Samantha" becomes "Sammy" or "Patricia" becomes "Patty" (Yeah I know the nicknames all sound similar but ending on a higher pitch AKA 'Y/I' lends itself to a level of cuteness).
Because of the shyness factor we should probably aim for less syllables in the name, so that she can say as little as possible to avoid prolonged social Interactions (Not that that will help her in something like the Birdhouse universe), so, maybe a Joan (Shortened to Joe), or Lou (Probably shortened from Lucy, the caretaker could probably call her Lulu in that case).
Those are just some examples and I'm sure you can come up with a solid name though.

Those are some good thoughts, thank you.

Susan!