Oh, man, if you did a search on Fanny Packs, would you get an eyefull!
I absolutely LOVE my fanny pack! I have a regular sized one and then I have my Theme Park Commando size, which holds a water bottle on each side, my Misty Mate, my video camera, regular camera, sunglasses, lipstick, hair clips, Purell (a must!) a tiny spray bottle of Oust (just in case!) pen, money, driver's license & cash card....Stuff that you really mentioned. When I use my regular sized one, I just hold my camera and my DH puts the video camera into the backpack that he carries (that also carries those cheap plastic ponchos we get at KMart for $1.99 and flip flops in case it rains.)
Oh, I forgot to mention: I carry my own pharmacy: Tums, Imodium, Tylenol & Tylenol Jr. band aids in a small plastic baggie.
My kids have had their own fanny pack since they were about 6 or 7. We would wait until we were down there to buy them one (they got one each trip) because they have special ones for kids that will fit their tiny waists.
They carry their lip balm, sunglasses, some of their money or Disney Dollars, small portable sunscreen, Purell, a pen and their Autograph Book (when they were younger) their disposeable camera (I told them "remember, don't wind it after you take a picture, wind it before you take your next picture so there's no photos of the inside of your fanny pack"). You daughter will feel very responsible being able to carry around her own stuff without having to worry about losing it!
You're gonna be out there all day and you don't want an unexpected trip back to the hotel/timeshare.
The awesome aspect of a fanny pack is that your hands won't get tired (I have carpal tunnel syndrome, so I can't carry stuff for long) and you won't forget it anywhere, since it's strapped around your waist. When you use the ladies room, you take it off and put it right back on. On some of the rides, you might have to move it to the side, but I've been using them for years and there is not one drawback of wearing a fanny pack versus holding a bag.
Most people will say it looks nerdy, but who gives

you know? I'm trying to have a great time without being weighed down, and the fanny pack is the best solution!
On one of our vacations together, my BF went and bought a fanny pack. She was excited to show me, but then when I saw her, it was bulging and looked like it would bust. I said, what the heck do you have in that thing? She pulls out her Big Full Wallet! Then a Huge Hairbrush!

Oh, it was so funny!!!
Here's me with my Commando Fanny Pack