I found out a little more about my baby's name.
"Xi" means 'tin'. Wuxi, the name of the city where his orphanage is, means 'without tin'. I don't know why. However, it is apparently represented using the characters for 'without' and 'tin'.
"Yun" means 'cloud'. In Chinese idiom, a cloud is something high above the common populace, something free - untethered and uncontrolled by terrestrial concerns. Much like a bird is free in English-speaking idiom.
"Gui" means 'noble, precious, valuable, expensive'. This is another name that sets my son above and apart from everyone else. It's a reasonably common given name, not surprisingly. 'Yun' is common as well. I've heard that in China and much of Asia, a child's name is supposed to be fortuitous, and of course everyone wants their child to rise above everyone else's children.
'Xi' is a rare and unusual surname. Maybe it will grow more common due to the practice of this orphanage of gifting its children with the surname. It will be an interesting note in the study of genealogy that people surnamed 'tin' will be able to trace their family back to the orphanage.
The whole thing is pronounced "sure young guay." I had been mistakenly saying, "she young gwee." I still need practice deciphering pinyin, but I'm learning.
By the way, my source was thirdhand, so I can't vouch for any of this. A friend, whose parents are from Taiwan, asked her friend, who grew up in Wuxi. I'd be inclined to trust its accuracy, at least as a starting point if we ever decide to research Yun Gui's history in greater depth. But I'm also kind of lazy and haven't made any plans to start the aforementioned research.