Is there a word to describe when a person with a non-English name adopts an English name, because it happens to sound very much like her real name?

Anglicisation (more commonly spelled as anglicization) is often used. It refers to "the process by which something or someone (usually a word) is made more English."

Some other processes possibly related to such name changes are

• acculturation, "A process by which a person acquires the culture of the society that he/she inhabits", from acculturate, "To change one's culture based on the influence of another culture"
• euphemization, using "a word or phrase to replace another with one that is considered less offensive or less vulgar than the word or phrase it replaces"
• accommodation, a state of being fitted and adapted; "An adjustment of differences; state of agreement; reconciliation; settlement" (sense 5); "The application of a writer's language, on the ground of analogy, to something not originally referred to or intended" (sense 6)
• normalization, "Any process that makes something more normal or regular, which typically means conforming to some regularity or rule, or returning from some state of abnormality"
• romanisation, "Putting text into the Latin (Roman) alphabet"; may refer to simple transcription or transliteration of names
• assimilation, "The adoption, by a minority group, of the customs and attitudes of the dominant culture"

These are cases of deriving a new name homophonically (or homophonetically, if you want to mash up homophone and phonetic) -- taking advantage, as you say, of the fact that "Lu Xi" sounds approximately like "Lucy." (I knew a girl named "Ka-Lin" who became "Catalina" by the same method...)