The correct option is
C Either
Option A - 'Neither' is a conjunction which means none. For example, 'Neither of you have won the game.' 'Neither'is often used with 'nor'. For example, 'Neither you nor Harry has won the game.' As the given sentence has 'or' we can't use 'neither' here. Thus option A is incorrect.
Option C - 'For' is used to indicate the duration of an action. For example, 'I have worked for two hours.' As the given sentence doesn't define the duration, we can't use 'for' here. Thus option C is incorrect.
Option D - 'If' is a conditional conjucntion, which is used to explain a situation's consequences. For example, 'If I were rich, I would have bought a lot of gold.' 'If' is often used with 'then'. As the given sentence doesn't explain conditions, we can't use 'if' here. Thus option D is incorrect.
Option B - 'Either' is used as 'any'. For example, 'Either of you can buy the dress.' It is often used with 'or' to explain choices in a sentence. As the given sentence has 'or' and has choices 'he or his brother', we can use 'either' here. Thus option B is the correct answer.
'Either he or his brother has sung this melodious song.'