The key goals of this exchange seem to be 1) matching the correct pronouns and 2) identifying the more important verb "invite".
We thought about inviting Lucy to come with us.
This statement includes both pronouns and verbs the listener must understand.
1) Did she think so?
"She" is the correct singular-female pronoun to refer to "Lucy," but this responses focuses on the wrong verb - "think/thought" instead of "invite." It's a distractor that references the words used in the original statement, but in an incorrect way. The statement isn't about what Lucy thought, so this answer doesn't make sense.
2) Why did she?
Why did who do what? "She" in this case can only refer to Lucy. There's nothing in the statement to indicate Lucy did anything. The people who did something are the "we" referred to by the speaker. Since the pronouns are incompatible and referring to the wrong person this response is incorrect.
3) Did they accept?
This response focuses on the proper verb: "invite." It requires that the listener recognize that the verb "accept" is used to respond positively to an invitation. It's incorrect, however, because "they" is a pronoun for a group, not for a single person, so it cannot refer to Lucy. "Did she accept?" would be ok.
4) Why didn't you?
As you noted, this is the correct answer. "You" in English can refer to a single person (the speaker) or a group (the "we" mentioned by the speaker). The sentence indicates that they thought about inviting Lucy, but they did not (otherwise they would likely say "we invited Lucy to come with us). The complete idea is "Why didn't you [invite her]?"