The above is my Chinese name, given to me by my first Chinese language teacher at the University of Toronto. If you research anything China related you most likely have a Chinese name that somehow-kinda-sorta sounds like either your first name, last name, or both (depending on the number of syllables).
In my case, 'Sarkissian' is actually three syllables long (pronounced sar-KEESS-yahn, though most people understandably pronounce it sar-kiss-ee-an). The three syllables match the length of the most common, three-character Chinese name. But there is no way to say ‘sar-KEESS-yahn’ in Chinese. The set of possible syllables in Mandarin is very small, and in practice there are only a few hundred distinct syllables, so it's tough getting a good match in pronunciation. The closest approximation in my case is shā-qĭ-shàn (pronounced roughly 'shah-chee-shahn').
Surnames carry no meanings over from the character's other uses, and are best seen as proper names. My given surname Shā 沙 is very peculiar and rare. The only famous figure in Chinese history with 沙 as a surname (so far as I know) is 沙悟凈 - Shā Wùjìng, a rather gnarly looking (and, coincidentally, bald!) disciple of the Buddhist monk Xuánzàng in the epic novel Journey to the West (written ca. 1590s).
Personal names do, however, carry meaning. My personal name is perhaps equally strange, but it also (strangely!) reflects much of my academic research.
qĭ 啟 = revealing; shàn 善 = goodness