The list of states that are the most well-known to foreigners and receive the most visitors would probably be: (In alphabetical order)
Those would be the big five. California is the most populous US state, it's where you can find Los Angeles and Hollywood, San Francisco, Yosemite and many other places, so it sees a lot of visitors. Florida is home to Disney World, a host of other theme parks and plenty of beaches, so it's also very popular. Hawaii has a great climate, spectacular scenery and it's the closest part of the US for people traveling from places like Korea and Japan and Australia, so it's popular. Nevada has Las Vegas, which is obviously the biggest draw by far, but it's also got Reno and Lake Tahoe, so although it doesn't seem as though it's got a whole lot going for it, it's definitely one of the most visited US states. Then of course there's New York, where you can find New York City, a place that gets heaps of tourists from all over the world.
Considering that you're asking for an "underrated" state, I'm not going to list states that are only known for one particular thing or one particular area. Examples would include:
Alaska - everybody knows that it's stunningly beautiful and a very unique part of the country.
Texas - everything is bigger in Texas, lots of history and character, but definitely not off the beaten trek by any means.
Idaho, Montana, and Wyoming - plenty of nature, the national parks receive a lot of visitors, but not exactly hidden gems either.
A similar argument could be made for Colorado, Utah and of course Arizona, because millions upon millions of people go there every year to see the Grand Canyon.
Here are my top picks:
North Carolina. It's a state that's got an incredible amount of variety to it. It's got world-class beaches, the mountain scenery is just breathtaking. Standing atop a grassy bald in the Great Smoky Mountains is an incredible experience. There's surfing, skiing, hiking, there's a lot of history, and there are some really interesting places. Asheville is very trendy and artsy. If you could combine your visit to North Carolina with a side trip to South Carolina you could see Charleston as well, which is a really cool city and you could tuck into some shrimp and grits and stroll around. The Carolinas are actually a great part of the country.
My next pick would be Michigan. A lot of people, even Americans themselves, don't know that Michigan has the one of the longest coastlines of any US state. Detroit isn't a great city, but it certainly used to be. There's a lot of history there. And Michigan has a lot of great outdoorsy stuff to do - boating, camping, fishing, hiking, that sort of thing. You could tour Detroit - it's nowhere near as bad as it used to be today, see a lot of museums reflecting the states industrial past, lots of lakeside mansions and other cool stuff, head up to Mackinac Island or all the way up to the Upper Peninsula where it's just pristine. The people are cool, it's got its own little vibe. And you can easily cross into Canada for day tripping there as well from places like Detroit or Sault Ste. Marie. Definitely worth a visit.
New Mexico. It's a huge state, and there's a lot to see and do. Eat great Mexican food, head to Clovis and see the prehistoric sites, check out the museums celebrating the area's contributions to rock and roll music. Sante Fe and Taos have great art scenes. The scenery in New Mexico is great, the mountains, the desert, it's got a lot to offer but most people just see it as a blank space between Texas and the West Coast. Many go straight through it on their way to the Grand Canyon in Arizona and don't stop to take advantage of what's there. There's Carlsbad Caverns, the nuclear testing sites, Roswell, hot springs, white water rafting, old Spanish missions, heaps of Native American attractions. Well worth a visit.
There are a few other areas I'd recommend, but they're fairly concentrated in only one part of the state so I'm not listing the states where they can be found because there's not much else:
The Massachusetts seacoast - Cape Cod, New Bedford whaling museum, Boston, etc.
The Ohio Valley - parts of Kentucky, West Virginia and Ohio, but other than beautiful scenery and nature, not much else.
New Orleans. A very unique part of the USA. But the rest of Louisiana isn't that great.