What are some easy, toddler-friendly recipes for a holiday party?

Quesadillas, you can hide lots of healthy things in there.
Mini chicken cordon bleu, these are great for adults too.
Mini meatballs
Blanched veggies with dip, many toddlers have a hard time with raw veg
Fruit kabobs with yogurt dip, healthy dessert
Japanese style veggie pancakes, again, fully of healthy stuff all hidden in a favorite food.

All of these are good for adults and can be made ahead if time and warmed at party time.

mini-meatballs are fun and often enjoyed. yogurt sundaes are v popular with my boys: yogurt/fruit/granola type of thing...

I highly recommend drbabs's cheese crispettes recipe http://food52.com/recipes/3112_cheese_crispettes. I made these for my daughter's birthday party last year and they were a huge hit with all of the kids.

Anything skewer-like is an option. As simple as olives on a toothpick (or even cut straw), cheese and grapes or banana and strawberries. Another option could be chicken strips for dipping (kids LOVE to dip just like lloreen said)

Here's a fun recipe for pizzas in muffin tins
http://www.marthastewart.com/360627/kid-friendly-vegetarian-recipes/@center/852566/vegetarian-recipes#284446

To lloreen's great ideas, I would add chicken fingers/nuggets.

Pigs in a blanket? Cucumber and cream cheese tea sandwiches (crusts removed, naturally). little triangles of grilled cheese sandwiches (toddler crack)
Toddlers generally like to dip things - maybe carrots and red pepper hummus-- sliced apples and greek yoghurt with cinnamon and vanilla. Anything where the ingredients are easily identified and don't touch.