In a book called Bible Basics - A Study Manual by the Christadelphian Bible Mission, it's stated that the gift of the Holy Spirit means the gift FROM the Holy Spirit, not necessarily the Spirit being the gift. In English, that could be possible, like we say, "..that blue shirt of yours" or "that huge TV of his". I can't think of when we would use that sentence construction for gifts though. English speakers say "gift of $300" or "gifts of food and clothing" or some such thing, referring to the gift itself and not the one from whom it came.
In the Hebrew before the translation to English (or maybe the translation from Greek to English), in what grammatical form is the word Spirit? What would that even be in English, object of the preposition? Possessive? It matters, doesn't it? But because the spirit is spoken of as being in us, working through us, I still think it means the spirit is the gift itself.