I'll second the quarters and pennies idea. When our kids were young, we got the kids tubes of mini-M&Ms before the trip and used those tubes to put quarters and pennies inside. I think they enjoyed the "squishy penny" hunts as much as anything else they got that trip. You can get folders that hold the pennies, so you can save them in one place.
As far as other souvenirs go, we stuck more along the line of what they could get, rather than how much we were going to spend, but we still had a budget in our mind. On our first trip, the kids were allowed to pick a pin for each park, then a pin, usually with the year on it, to remember that trip. Then they each picked a t-shirt and they were allowed to pick one slightly bigger item, that they waited until the end of the week to pick (so they could look at everything and decide). Since that trip, we've done the year pin and then a t-shirt or something similar.
I like the idea someone suggested to give them coupons. If they like using their own credit card, they would get a similar experience by handing over their "gift card" for one t-shirt, etc.