Here’s some of my faves:
Jade
Aralynn
Dakota
Morgan
Aurora
Isolde
Guenevere
Gwyneth
Musette
Musetta
Aurore
Arianna
Laura
Claudia
Niobe
Antonia
A name beginning with a vowel seems to be euphonius, flowing with the end of “Elizabeth”. But I’m coming up short on those.
Can’t go wrong picking from the names on this list. I especially like Magot. Actually if I were to suddenly up and spawn female progeny I’d be tempted to name her Idony or Juicia but everyone would think I made them up.
Helga
Edna
Maude
Bertha
Winnifred
Lucrecia
Betty
Joan
Helen
Mabel
Hagar
Madge
Flo
Hilda
Margo, yes, Margot, yes, but Magot?! Oh, no! Everyone would double the ‘g’ and her life wouldn’t be worth living.
Please, keep this serious. It is my daughter’s destiny we are talking about.
Check the vowels in your last name, and match them with the first name. That makes the names flow together well.
For example: Kimberly Miller. Or Kathryn Hansen.
If your last name really were Thornhill, it could be something like
Corinne Thornhill, or Morgan Thornhill (the unaccented a reduces to a schwa), Lorissa Thornhill.
It’ll help you to narrow things down, if nothing else.
Anglo-saxon names are serious business.
Theodosia
Hialeah
Lahaina
Lyonessa
Tarina
Absinthe (as in Absinthe makes the heart grow fonder)
Chickamauga
Tatra
Genoa
Alemagne
or a great one from one of Jack Vance’s sci-fi books…
…Alusz Ephegenia Eperje-Tokay
And she’s okay with that, right?
I didn’t realize this, I thought you were just giving her a middle name. In that case, I really like the following:
Rhiannon Elizabeth (very regal…Welsh goddess/Queen of England)
Elizabeth Anne (my own name, which I love)
Victoria Elizabeth (a myriad of nick names in that one)
laugh Think of all the time it would’ve saved if my parents had been that forward-thinking! 
With T & D in the last name, no T’s in the middle, it will sound like a stutter. Especially with Natalie.
I’d go with something one syllable (possibly two), to offset the 3 syllable first name and long last name, starting with a consonant sound, & ending with a consonant sound, being careful of l’s and n’s (start getting weird alliteration things if they’re too heavily there). And you want want the initials to work.
I like Natalie Claire or Natalie Beth (not Elizabeth with Natalie). I also kind of like “Jane” “Christine” “Grace” “Rose”
This is my favorite so far. I just enjoy saying it.
[sub]Chickamauga, Chickamauga, Chickamauga …[/sub]
Personally, I love this one, because it’s mentioned in Don Henley’s masterpiece “The Last Resort”. (“And you can leave it all behind and sail to Lahaina, Just like the missionaries did, so many years ago”)
But Natalie does not, so that’s that. She has just been through the thread with me, and her favourite so far is Louise, with Cordelia and Veronica close seconds.
Thank you so very very much. It means a lot to all of us.
Hmmm, I have two Canon Powershots. So yours is small. silver…how about “Sliver”?
If my parents had had a girl, they would have named her Louisa (“Lucy”). So please don’t let that dissuade you.
I used to dream of a woman named Elouisia, but I have never found anyone by that name. (It could have been spelled Alouisia or Eloisia, I guess.)
Tell Natalie that if she chooses Louise, she will have to take the middle name Ianna so that she will be named after the state where New Orleans is. 
Or maybe we should change her name to Katrina Rita. Did she behave in Paris? I’ll bet she liked the boats.
Anna Grace
Rosamond
Anna Clair
Rose Grace
Geez Louise…
Ella Rose
Zoe-Chloe
So much food for thought… As for Paris, her favourite bit was the fun fair set up near Le Louvre. She DID enjoy the van Goghs in the Orsay Museum, though. Like me, she was surprised how small the canvases were. As for the Batobus, it’s a wonderful way to get around. It poured with rain when we were on board, and she enjoyed watching the deck hand clamber on top and pull up the shelter sections one by one with a long pole. We were surprised it wasn’t automated. (And the Pole was pretty cheesed off, I can tell you…)
She enjoyed the cuisine too, though personally I would incline to believe that steak and fries tastes the same whether you’re in Burgundy or Gascony. Perhaps her palate is discerning enough to tell a Charolais from a Limousin bifsteak, though how that might be with the amount of ketchup she uses is beyond me.
Just a word about destiny…
take great care with the initials of her new name. Think how things panned out for poor Bill Jefferson Clinton