How would you express: "I have ________ people's names when I meet them"?

The phrase "I have a poor memory for people's names when I meet them" effectively communicates the idea that remembering names is not something the speaker is good at. This expression is commonly used to describe the difficulty one has in retaining information, particularly names, which is a frequent social challenge. The use of "poor memory" clearly indicates an inability or struggle to recall names, which aligns perfectly with the context of the statement about meeting people.

In contrast, the other options do not convey the intended meaning as accurately. "A troublesome memory for" suggests that the act of remembering names is problematic but lacks the specific indication of ineffectiveness. "A challenging memory of" implies a struggle but does not specifically relate to the act of remembering names. "A difficult memory with" also does not fit well as it sounds awkward and does not directly address the issue of remembering names. Thus, "a poor memory for" is the most suitable and commonly understood expression for the context provided.