This sentence can be checked with two quick rules: which words "either" and "neither" are allowed to pair with, and whether the articles before the two flights are complete.
The pairing rule is fixed: "either" always goes with "or" to present a positive choice between two things, while "neither" always goes with "nor" to rule both things out. Since the sentence means the speaker can take one of the two flights, a positive choice, it needs "either...or," never "either...nor" or "neither...or."
The article rule is about the two flights themselves. "The morning flight" and "the one in the afternoon" both refer to specific, already known flights, so each needs "the" in front of it. Dropping "the" before "morning flight," as in "either morning flight," leaves it sounding like any morning flight rather than the specific one meant.
Putting the two rules together, the sentence needs "either" paired with "or," and "the" before both flights.
So, the correct answer is We can take either the morning flight or the one in the afternoon.
Reading these four options aloud one at a time makes the odd one in each quickly noticeable.
The reading-aloud test flags a different snag in each of the first three options, leaving one sentence that sounds complete and natural from beginning to end.
So, the correct answer is We can take either the morning flight or the one in the afternoon.