Another question really got me to thinking about the Holy Spirit. I thought it better to open a new question rather to divert the focus of that thread. So growing up, I also thought of the 1st Person of the Trinity as Father and the 2nd Person of the Trinity as Son but never really came away with the overwhelming feeling of Holy Spirit as He. "It" would certainly be quite wrong, after all the Holy Spirit is the 3rd Person of the Trinity. What support is there for referring to the Holy Spirit as He?
There is never any indication in the New Testament that the Holy Spirit is a "he". In Greek, gender for normal nouns like "spirit", "rock" or "bread" is a grammatical thing, not a sex thing. This is common in the romance languages as well, if you are familiar with those. The fact that it is neuter does not tell you that it is an "it" as we think in English. The word "day" is feminine, but there is nothing feminine about the daytime. "Death" is masculine, but both males and females die. "teknon", one of the Greek words for child, is neuter, etc... So the fact that the word "spirit" is neuter (and it is, despite what was said in one of the other answers) actually doesn't tell you anything.
People often appeal to things like John 14:16 and 16:7 to prove that the Holy Spirit is male, but this is inaccurate. The word "helper" is masculine grammatically (like the words above). This is why pronouns that refer to "helper" like the "auton" in 16:7 are masculine. To do otherwise would be to commit a grammatical foul.
If you wish to say that the Holy Spirit is a "he", realize that this is an English thing you are doing, not a Greek. Orthodoxy states that the Holy Spirit is a person, so speaking of him/her/it personally entirely makes sense and is appropriate, but personhood in the Godhead need not imply gender. So, theologically, feel free to call him a person. But by arguing about gender you are taking this question beyond what the writers of the New Testament apparently felt like discussing.
Edit 9/11/2015 in response to comments because the response is too long to put in the comments themselves
@bruisedreed I can see where that could be confusing. Let's change the verse to something non-controversial to see my point. In John 19:23, it refers to Jesus' tunic with the statement ἦν δὲ ὁ χιτὼν ἄραφος..., translated by the ESV as "But the tunic was seamless". In the next verse it says μὴ σχίσωμεν αὐτόν... "let us not tear it". Grammatically speaking, αὐτόν/"auton" in John 19:24 is just as masculine as the αὐτόν/"auton" in the case we're discussing. Yet there is no debate on the personhood of the cloak and people would question the sanity of the ESV translators if they translated that last bit "let us not tear him". Unless it's one of those cases where we used a gendered pronoun when it isn't actually necessary but is idiomatically okay (e.g., "My boat, she's a sinking!"), in English we would generally choose "it" if something doesn't have an obvious gender and/or agency. So here "it" is appropriate because it would be weird in English to call a cloak "him". But in Greek that's not the case. If someone said μὴ σχίσωμεν αὐτό in reference to the cloak, he/she might be chastised for bad grammar because using the masculine is normal for the language in this case. John 16:7 is parallel. Greek has the pronoun as masculine because the noun it refers to is grammatically masculine. That in and of itself implies no gender or even agency. Jesus (or his translator into Greek) essentially had no more choice in the gender of the pronoun than whoever it was that said "let us not tear it (masculine pronoun)."
Now if παράκλητος/parakletos implies gender or agency, that's something else. That is meaningful. It clearly implies agency but it is debatable that it implies gender.
@warren you say "so saying it's merely an English thing to assign gender to persons of the Godhead is false". I agree. However, what I said was "need not imply gender", not "does not imply gender". Perhaps I should have said "doesn't necessarily imply gender".
@Susan That's really interesting. So here's the question I have about this. Idiomatically speaking, when referring to an agent and substantized, is this adjective/noun always masculine? If it is, then there isn't really a choice that can be made here (grammatically speaking) and the "choice" of gender isn't a choice at all and just what is expected in the language. If it isn't, then a masculine παράκλητος here might be useful for arguing gender, which could be useful for those concerned with giving the Spirit a gender. I was hoping the Philo reference in De opificio mundi would be helpful but I don't think so :( I don't know the answer to this but I think it's an interesting question. More usages of the word would be helpful. The form you mention clearly shows that there can be a feminine (in both grammatical and sex senses) παρακλήτρια. But is there enough data to prove that παράκλητος is a dude? Somebody get us all the references to the term in Greek literature and we'll figure it out :)
Jesus calls the Holy Spirit "him":