I am traveling to China for 10 days in March, and want to add a few days to hit another country. Suggestions?

Any suggestions for another country/city to hit while I am over there - somewhere in Thailand? Tokyo? Hong Kong? I would be flying out of Beijing.

Hongkong and Taiwan.
Certainly not Thailand and Tokyo. 10 days won't suffice.

Do Thailand along with the other south east asian countries.
Or Tokyo with Korea.
But none of them should be visited in a 10-day period including China.

10 days is not enough time for China but, I might stop Tokyo because it is so different and a world class experience. Hong Kong is impressive, too, but, for me it is not as exciting as Tokyo.

10 days is really not much time in China unless you are spending it all in Beijing. Inside China, I'd suggest checking out Xi'an, Xiamen, Yunnan, Xin Jiang, Tibet or Sichaun. None the less, if you want another destination outside of China, I hear great things about the beaches in Thailand. ...and everything there is dirt cheap. Also the cherry blossoms maybe blooming in Kyoto, Japan. Have a great trip!

It depends on your expectation.
Inside China, Guilin and SuZhou for sightseeing and Hong Kong for shopping and nightlife.
Outside China, yes Tokyo, I love it. Kenting in Taiwan is also recommended for relaxing.

Traveling to China and visit other countries will be a very tough one as China is so big and its really up to you on whether where you want to go. For 10 days, visit Shanghai, Suzhou and Hangzhou or Nanjing will be suffice. As you are flying out of Beijing, travel to Tokyo will be nearer than the rest of the place you quote above. You can also try Seoul or Pusan as its even nearer. If you spend too much time in Beijing, Hong Kong is another bet as you will only need 3 days out of 10 days to visit the major attractions in Hong Kong. If you want, Manila is another place for you to consider if you want to visit a nearby country for a short period of time. Many flights flies out of Hong Kong to Manila daily and its only take about 2 hours to get you there.

This are some suggestion for you:
Beijing (7days), Hong Kong (3 days) or Beijing (5 days), Tokyo (3 days), Osaka (2 days) or Beijing (6 days), Osaka (4 days) or Beijing (5 days), Tokyo (5 days) or Beijing (5 days), Hong Kong (3 days), Manila (2 days) or Beijing (6 days), Seoul or Pusan (4 days) etc.

Hope the above can solve your travel problem, have a nice trip ahead of you.

MONGOLIA