Every boy and girl got the present;____ was given on New Year's Eve.
a.she b. he c. they d. it

Since there are only these choices, only "it" can be the answer. The semi-colon here suggests that further information is following. Clearly that information can not be that a he or a she was given. If so, you would have to ask given what or to whom. Obviously, "they was" won't work either.

So: Every boy and girl got the present; it was given (to them [understood]) on New Year's Eve.

However, in this situation I would explain that the sentence itself, though not grammatically incorrect, is bothersome and could be written far more clearly without changing the meaning. Part of the problem with this sentence is that we have to assume a lot about "the present." Is this sentence meant to say "a present?" If not, we can be led to believe that there is a present that each boy and girl already knew about.

All in all, I'd probably end up writing the sentence like this: On New Year's Eve, every boy and girl got a present.

However, if it is to be implied that the children were given a specific present that they already knew about or that the writer of the sentence already knew about and assumed the reader knew about, I would write it this way: The present was given to every boy and girl on New Year's Eve.

No matter what, I would not expect to find this sentence in a proper textbook for EFL students. I would imagine that it was written by a Chinese, English grammar teacher. But that's just a guess.