There are quite a number of options, although the 4000 foot requirement pretty much limits it to the White Mountains of NH, the northern Green Mountains of VT, and a few peaks in ME.
It's not clear if you want to or need to do a out and back, or if you can spot a car and do a traverse. Here are just a few options that come to mind without looking things up on a map:
- From the end of Sawyer River Road in Crawford Notch, hike up Carrigain one day, stay the night, and back down the next. There can be some good views, although I don't remember now if the old fire tower on top has been taken down or not. There are great views from there if it's still there. This is fairly easy for the alloted time. You can arrive in afternoon on the first day without it causing a problem.
I don't know about waterfalls, but there will be some stream crossings before you get onto Signal Ridge.
- Take the Ethan Pond Trail from Crawford Notch to Thoreau Falls. From there you can go down by the falls, south, then west, then north again over Bondcliff, Bond, Guyot, Zealand, past Zealand Falls Hut, and back to Crawford Notch. This is quite ambitious, but doable if you start early in the morning and are in good shape and willing to do a lot of hiking for the alloted time. You'd probably spend the night partway up Bondcliff.
The upside is you go by several falls, and the top of Bondcliff is probably the most spectacular place to stand in the White Mountains. I screwed up once and ended up spending the night on Bondcliff. I don't recommend that. Plan better than I did.
If this still isn't enough of a challenge, you take the Zeacliff trail "shortcut" down from Zealand Mt instead of going by the hut. I did that once too, and decided it should be a up-only trail, so I also don't recommend that.
- Go to some of the waterfalls on the west side of Crawford Notch, but keep going instead of back to the parking lot like everyone else. There are ways to loop around and come out a different place in Crawford Notch. There should be at least a couple of choices here you can find by studying a map. I don't have one in front of me, so can't say much more about this off the top of my head.
- There are various options for one-night backpacks in the Carter-Moriah range east of Pinkham Notch. For example, start going up the Nineteen Mile Brook Trail, then pick a way that loops back to the road eventually. There should be three basic choices, with the southern one including Carter Dome.
There are lots more possibilities. Get a map and look. However, one thing you probably don't want to do is Franconia Ridge. There are some pretty waterfalls on the Falling Waters trail, but there are no good places to camp on top. Either it's above tree line or within a exclusion zone, pretty much all the way from Liberty to Garfield. This is also the most crowded backcountry area in the White Mountains because of the number of trails providing access from Franconia Notch, and the presence of Greenleaf Hut.
If you're looking for a ambitious but certainly doable day hike, then this works better. One possibility is up Falling Waters, north along the ridge over Lincoln, to LaFayette, down to the hut, and back to the same parking lot via the Bridle Path. You get great views for quite a ways along Franconia Ridge, which is above treeline from Haystack to just above the hut on this route. If you do this, bring winter clothes in your pack. You won't need crampons in the summer, but a good rain/wind shell, wind pants, extra wool sweater to put under the rain shell, goggles, and gloves is NOT overdoing it. The weather can change there very quickly on short notice. I've started out in shorts and a T shirt from Liberty Springs in the morning, and was wearing full winter gear except for crampons by the time I got to LaFayette. At that time 20 mile/hour sleet was blowing in my face, and I was very glad to have goggles on. This was in August. Yes, it can do that, and you can't see it coming due to Kinsman Ridge on the other side of Franconia Notch.