We're birthday visitors due to our daughter - our tradition has been to visit DURING her birthday since she turned one. So, I've researched the heck out of this subject to make sure we get it all! I say do it during your entire stay to get the entire experience for the birthday child!
Disneyland is the best place for birthdays! Make sure to get a birthday button (at your hotel or City Hall on main street.) You can arrange a birthday phone call as well if you're staying at a Disney resort (just ask the hotel.) If not a resort guest, go to City Hall and tell them you have a birthday and want to hear the phone call.
Also if you're staying at a resort hotel, you can arrange to have balloons and autographed pics of the favorite characters waiting - we did through Vacation Planning but I believe you can also ask the hotel to arrange it. It's free.
There is a My Disneyland Birthday Party you can make a reservation for - anyone can do it and it's generally daily at 2pm (check with VP or Disney Dining for the day you want it.) Held at Plaza Inn and it's $10 per person. Everyone has to play but it's well worth the price. Everyone gets to decorate their own cake which comes in a "toodles" shaped container with the cake at the head and toodles sprinkles, m&m type candies, and several colors of frosting in the ears. Also a cute plastic cup with plastic straw filled with fruit punch is yours to keep! Stickers are handed out like wild fire and in addition to the zany host leading activities, Mickey and Minnie make an appearance as well. You will get to dance and have your pictures taken with Mickey and Minnie. This party is well worth it and a MUST DO for birthdays! I can't recommend this enough, was successful in talking another board member to do it on her last trip and she raved as well!!!
You might also want to do Build A Bear (they have special early hours on certain days of the week for resort guests - arrange through Vacation Planning) as they are the only location that has licensed Disney clothing.
This isn't really something for you, but in case someone else finds this looking for ideas - Libby Lu is something girls also really like (if you go to their website you can put her in the VIP program for free and get a discount on anything you get there, you just fill out the paper and take it with you.) Girls can get a "make-over" and even make her own sparkly lotion or lip gloss. Really neat girlie stuff!!! She'll have a blast being catered to - make an appointment through Libby Lu for that too so she feels special and has no waiting. I will admit to having a sparkly lotion of my own...
Priority seating for Playhouse Disney is another option, Vacation Planning also does that! In fact, Priority Seating is nice for other things like Fantasmic and Electric Parade. Extra shows for birthdays are always nice, especially for the older crowd.
You might think about making a reservation at Carnation Cafe on Main Street - at least for a dessert for your special birthday! You can get the Birthday Bags and Buckets at Carnation as well!
Character meals are wonderful for birthdays and the staff really goes the extra mile to attend to their birthday guests - Goofy's Kitchen (breakfast, lunch, dinner), Ariel's Grotto (lunch, dinner), Storyteller's Cafe (breakfast), Lilo & Stitch (breakfast), and Plaza Inn (breakfast.) Reservations can be made 60 days in advance. At these locations, you can order a birthday bag $5 (puzzle, Goofy pen, maraca, etc.) or birthday bucket $20 (small plush of Sorcerer Mickey was in our Mickey-themed bucket, along with a collectible Happy Birthday pin, and an adorable cake with Mickey on it - it was beautiful and easily 2 servings.)
Think about baskets and sets from Vacation Planning, They are more expensive, but it's not bad and you do get high quality items. Several years ago I got a basket as a gift on a trip and it was incredible!!! We also got one for Morgan on her last two birthday trips and they tailored it especially for her with crayons, a coloring book, and lots of toddler-friendly candy. This staff also goes the extra mile to help out and attend to any special needs - like avoiding candy that could cause choking.
Other Vacation Planning Items that we've enjoyed are Mickey Cookies & Milk or Mickey Rice Crispies and Milk: $25.65 plus tax (19%), plus a 2.00 delivery fee. They are large so easily shared. If you don't want the milk you can order the cookies or rice krispie treats without milk for $2.50 each plus the tax and delivery charge. So, non milk drinkers - you might want to consider this option! You can also order chocolate milk or whole milk if you need this for your child. You can also arrange this through room service if needed at the last minute.
Don't forget to stop by the Mexican restaurant inside Disneyland as well, for your birthday you will receive a free package of fresh, homemade tortillas - a lovely snack!
Happy Birthday!