Keep 'em coming!
Marley, thanks for the clarification on the ones you mentioned.
racinchikki, I know that the first one that her mom mentioned as an example was one by the Dixie Chicks. She likes lots of kinds of music, but the D.C.s are a favorite.
“Cat’s in the Cradle” by Harry Chapin? 
“Walking on Sunshine” by Katrina and the Waves.
Leavin’ on a Jet Plane
The Letter
Time After Time
Bluer than Blue
Tears in Heaven
Not those. Don’t use those. Use these instead:
Wind Beneath My Wings (no, seriously!)
Leaving’s Not Leaving
There’s “Tenderness on the Block” by Warren Zevon:
“I know how much it hurts to see her go
She has a mind of her own, you know
She’s all grown up
She has a young man waiting
She’s all grown up
She has a young man waiting
She was wide-eyed
Now she’s street-wise
To the lies
And the jive talk
She’ll find true love
And tenderness on the block”
Full lyrics here.
When I left home it was years after moving out of my parents’ house but I hadn’t quite cut the cord because I kept going back. So, when I finally decided to move from Kauai to Oahu, it was to the tune of “Jet Airliner” by Steve Miller Band. And I haven’t looked back since. But, oh how I miss them. 
Oh yeah, I like Beautiful (Carole King) a lot, but it makes me cry. It doesn’t hurt to remind someone who’s stepping out on their own that You’ve Got a Friend (Carole King’s rendition also on Tapestry is, IMO, a little more upbeat than James Taylor’s). It’s especially nice to be reminded that your mommy is the bestest friend you could ever have.
My mom made me a mix tape when I left home. She included…
See Ya Later, Alligator - Bill Haley
Hit The Road, Jack - Ray Charles
Don’t Come Around Here No More - Tom Petty
Changed The Locks - Lucinda Williams
My Aim Is True - Elvis Costello
Sunshine on Leith - The Proclaimers
Van Morrison has a trio of good songs for the occasion (all on the Tupelo Honey CD):
Wild Night - Van Morrison (sort of an uplifting “seize the day” song)(or “seize the night,” I guess)
Starting a New Life - Van Morrison (like the title says)
Tupelo Honey - Van Morrison
(What a) Wonderful World - Louis Armstrong
Put the Message in the Box - World Party (This one is perfect for the OP’s purpose- very upbeat, and it’s about going out into the world.)
Blind Faith - Can’t Find My Way Home
What?
Oh. Never mind.
Gonna Move - Susan Tedeschi
One of my all-time favorite artistes…
And then I saw Marley23 had already mentioned it. Day late and $1 short…
“Do Re Mi” - Nanci Griffith, Other Voices, Other Rooms (This is not the Sound of Music song!)
“Listen to the Radio” - Nanci Griffith, Wings to Fly and a Place to Be
“Hammer and Nail” - Indigo Girls, Nomads, Indians, Saints
“I Feel Lucky” and “Bug”- Mary Chapin Carpenter, Come On, Come On
“I Be Your Water” - Sweet Honey in the Rock, In This Land
What a great idea, Zoe! She’s very lucky to have such a cool mom and grandma.
Zoe, I suggest you google for the lyrics on the songs you don’t know. Most of the suggestions above are good, but just a few strike me as wildly inappropriate for your purpose.
Just thought of another one, a bluegrass number (which seems appropriate for a North Carolina girl).
It’s the old Carter Family song entitled You Are My Flower. I recommend the version performed by Lester Flatt and Earl Scruggs on their greatest hits compilation 'Tis Sweet to Be Remembered.
*So wear a happy smile and life will be worthwhile
Forget your tears and don’t forget to smile
You are my flower…*
etc.
And then of course there’s These Are the Days by 10,000 Maniacs. Classic “seize the day” stuff.
Also You Are the Everything by R.E.M. (from the Green CD).
Memories of East Texas - Michele Shocked - Short Sharp Shocked
Teach your Children - CSNY - So Far
Sheryl Crow, “Soak up the Sun,” “Hole in My Pocket”
Beautiful Day U2
Walk On U2
a second for
You Gotta Be Des’ree
The Pascagoula Run Jimmy Buffet
That song from that Coke comercial that there was a thread about yesterday.
Shadowland The Lion King B’way soundtrack
Ok not exactly ‘happy’ but very powerful. If you are not familar with the broadway show, this song is sung by Nala and the rest of the women as she leaves the prideland because King Scar want her to bear his children and things in general suck.
And where the journey may lead you
let this prayer be your guide
though it may take you so far away
always remember your pride.
It’s a powerful song from the show.
OH and
Que Sara Sara by Doris Day
“Army” by Ben Folds Five (uses the ‘f’ word tho)
“Redneck Past” by Ben Folds Five
“These Boots Were Made for Walkin” by Nancy Sinatra