This isn't money for food, snack, or tickets to attractions, is it? We're just talking about souvenirs, right?
My vote probably isn't popular, but I suggest zero.
Yes, really. When our kids were toddlers we realized that souvenirs could essentially take over a trip. We want the trip to focus on the experience, on learning something or just having fun, on spending time with the family . . . but when the purchase of toys and such are involved, so often the kids become so excited about making their choices that they rush through the experience so they can get to the shopping. And then there were the times when the toy broke as soon as we got to the car, which was always a disaster. We tried limiting to one souvenir, but that was almost worse: With only one item to choose, the pressure to "get it right" was intense. We tried limiting by giving the kids money, but again, the problem of stuff overshadowing experience remained. And then what happens to the stuff when it gets home? The vast majority of it, although it was rather high-priced, doesn't become a much-loved item; rather, it tends to be tossed into the corners of their bedrooms and eventually end up in the Goodwill box.
So about the time our kids started school, we hit up on the NO SOUVENIRS concept. We let the kids know this new stand up front, and we encountered essentially no flack from them. They understand that they've been to more places than their friends, and they understand that we have not good but GREAT trips. They've learned to value that above bringing home yet another tee-shirt or stuffed animal.
Please understand that this isn't about the money. It's about maximizing the quality of your trip.
Since we made this decision, I've chaperoned numerous youth groups on many trips, and I've seen a positive difference in my kids on this subject. For example, not long ago I was a chaperone for a trip to the aquarium. The kids had a wonderful time looking at the fish, watching the guy feed the fish in one tank, calling each other over to see something great . . . until they neared the end of the experience, and the girls realized that there was a gift shop at the end of the aquarium. 50% of the kids instantly were DONE with the fish and hurried straight to the shopping, bypassing the last exhibits altogether. Most of the others gave the remaining exhibits only a cursory glance and set their sights on buying something. 2-3 kids out of the large group, my daughter being one of them, ignored the gift shop and continued to marvel over the last exhibits.
Another memorable experience was a several-day trip in which one of the girls in our group spent ALL her souvenir money at our very first stop, and then wailed about it long and loud when better options came available later. It's worth noting that our first stop, where she spent every penny, was a gas station.
I suggest that if you aim for NOTHING, you'll end up with just the right amount of stuff. You will see a little something here and there that's just perfect and unexpected, and you'll buy those things then.
If you're concerned about missing out on a learning opportunity dealing with money, that's a lesson that can be put into practice any week at home. Let them have some control over grocery purchases. Let them learn that store-brand this-or-that is the same as name brand. Give them X amount of money for their school supplies, and let them decide how to spend it. But don't trade your vacation for the accumulation of a bunch of trinkets.
Changing the subject slightly, we did a cross-country driving trip a few years ago, and it was wonderful. My husband and I agree it's the best vacation we ever took. Our girls were 15 and 12 at the time, and they didn't necessarily enjoy all the history at the time, but we interspersed it with plenty of water parks and other great experiences. Four years later, they still bring up things from that trip: Remember when we did the cookout ride, and Dad's horse bit Mom? Remember when we saw that great 4th of July parade at the Grand Canyon, and they sprayed us with water? Remember that hike in Grand Teton where we were so exhausted by the time we reached the top that we just laid on that big slab of granite for like an hour looking down over the lake, and we threw crumbs of our sandwiches to the chipmunks? Remember doing the buffalo safari?
What do they NOT bring up? Our lack of shopping for souvenirs.
I wish I could take an 8 week vacation. I hope it will be pleasant!
we will have space limitations too, in the rv with three kids and two dogs... So no giant teddy bears at least! Our youngest will have his birthday while on the trip, and our oldest will have hers shortly upon returning, so the kids will want to be able to buy things for each other as well. Mall of America is one of our stops so I'm sure that will pry some cash from the kid's hands 
