The correct option is
B neither
NOTE: 'Nor' is used to explain negative choices/options in a sentence. Hence, we use a negative conjunction to fill in the blank.
Option A - 'Either' means 'anyone of'. It is often used with 'or'. For example, 'Either Raj or Simran has the guitar.' As the given sentence doesn't have a positive choice, we can't use 'either'. Thus option A is incorrect.
Option C - 'While' means 'during the act/at the same time'. For example, 'I read it while you were working.' As the sentence requires a negative work to pair with 'nor', we can't use 'while' here. Thus option C is incorrect.
Option D - 'Both' means 'in addition to'. For example, 'I want both ice cream and chocolate.' As the given sentence explains 'none of the options' we can't use 'both' here. Thus option D is incorrect.
Option B - 'Neither' means 'none'. It is often used with 'nor'. For example, 'Neither Raj nor Simran has the guitar.' As the given sentence explains that none of the items is liked by the subject, we can use 'neither' here. Thus option B is the correct answer.
'I like neither tea nor coffee.'