This question is testing MODIFIERS.

If we have an "and" linking two nouns (e.g. "Latimer and Edison") followed by a noun modifier (e.g. "who"), it's implied that both nouns are being modified. So in "... Lewis Latimer and Thomas Edison, who became known...", it's implied that both men became known. But since that's followed by "his invention," we know that we should only modify one man, not both.

A. of Lewis Latimer and Thomas Edison, who became known for his invention of the light bulb,
- "who" is incorrectly modifying both men

B. of Lewis Latimer and Thomas Edison, known for his invention of the light bulb,
- switching to "known" doesn't fix the problem, because it's still a noun modifier incorrectly modifying both men

C. of Thomas Edison, known for his invention of the light bulb, and Lewis Latimer
- no issues - the modifier is correctly modifying just "Edison"

D. of Lewis Latimer and Thomas Edison became known for his invention of the light bulb and
- same issue with the modifier applying to both men
- the structure of the sentence is now "the research... became known for... and accelerated..." This is nonsensical. Edison was known for the lightbulb; the research accelerated development.

E. that was conducted by Thomas Edison, who became known for his invention of the light bulb, and Lewis Latimer
- unnecessarily wordy, though not strictly grammatically incorrect. This fixes the modifier issues of A, B, and D, but C gets the same information across more succinctly, without unnecessary tense makers such as "was conducted" and "who became known."

The correct answer is C.