There are a few campsites around Lake Dillon. Ten miles or so from Breck. Some will take reservations. Here's a website. Scroll down to "Dillon Ranger District"
US Forest Service, White River National Forest - Campgrounds
The campsites around Lake Dillion are going to be, um, non-primitive. Think RV's, dogs, paved parking, kids on scooters, etc. They will also fill up every weekend in the summer and many weekdays.
Don't know what your soon-to-be relatives are expecting. You might have a hard time finding anything closer to Breck. Lot's of private land there. Maybe a local could give you better direction. There's also this thing called the internet, maybe you could use that.
Wait, was that constructive?!