I have recently been inspired to reexamine this passage and have changed my thoughts a bit. I am leaving, rather than editing, my original answer since it has been around for several years.
A video version of this answer with some additional information can be found on my YouTube channel.
The Hebrew ר֣וּחַ (ruah) can mean spirit, breath, or wind based on context. The verb used in the clause in question, מְרַחֶ֖פֶת (root = rachaph), is quite rare occurring just three times in the Bible. As it turns out, determining its precise meaning will be the key to understanding Genesis 1:2.
Whenever faced with an interpretative difficulty, the first thing to do is check the immediate context of the verse. There isn't much in Genesis 1 to help us, although we do see “God” mentioned several times, which corresponds to the Hebrew elohim in every case.
The next thing I like to do is check how ancient translators rendered the verse. This is not a step a lot of people take, but it makes a lot of sense. The ancient translators are a lot closer to the text both in native speaking ability and in time than we are. As it turns out, spirit and wind are the same word in all the relevant languages, so we are just left with the verb to examine.
The oldest translation of the Hebrew Bible is the Greek Septuagint. Unfortunately it offers little help, choosing a rather generic verb meaning roughly “brought upon.” Next up are the Aramaic Targums. Targum Onkelos chooses “blow” in a way that suggests ruah is seen as wind. Targum Pseduo-Jonathan and Neofiti use the same verb, but an “of mercy” after ruah making it less clear if they are think of a merciful spirit breathing on the waters or a wind of mercy blowing on the waters. The Syraic Peshitta makes perhaps the most interesting chose opting for a verb thats main meaning is broods (as in an animal caring for its young). Finally, the Latin Vulgate chooses a verb meaning was bearing (as in children) or was carrying.
So we have some support for wind from God and some for Spirit of God from ancient translations. To resolve the wind vs. spirit dilemma, we'll have to continue pushing on the meaning of rachaph.
Indeed, it only occurs in two other places in the Bible. In Jeremiah 23:9, it means something like tremble. However, this is a different verb stem and so the meaning is not necessarily close to the usage in Genesis 1:2. Deuteronomy 32:11, however, is quite helpful. Starting at Deuteronomy 32:9 we read:
But the Lord’s portion is his people, Jacob his allotted heritage. He found him in a desert land, and in the howling waste of the wilderness; he encircled him, he cared for him, he kept him as the apple of his eye. Like an eagle that stirs up its nest, that (yerahep) over its young, spreading out its wings, catching them, bearing them on its pinions (ESV)
There was a time when scholars (and me in my original answer) thought 32:11 and possible Genesis 1:2 could be painting a picture of a brooding bird, in line with the Syriac cognate we saw in the Peshitta. However, this view has generally been discarded in light of the stronger Ugaritic cognate. Interestingly, the Ugaritic term is known only is the context of eagles where it appears to mean soar or hover. Either way, the picture is of a bird in flight.
Dead Sea Scroll 4Q521
Further support for the translation hovering comes from the Dead Sea Scrolls. As previously mentioned, as I previously mentioned, the piel stem of rachaph occurs just twice in scripture. It does, however, appear one other time in ancient Jewish writings, in a very interesting text, 4Q521, or more descriptive the Cave 4 Messianic Apocalypse. The text speaks of how the heavens and the earth will listen to the Messiah and through him will experience the Lord. It then remarks
“For the Lord will consider the pious, and call the righteous by name, and his spirit (ruah) will hover (rachaph) upon the poor, and he will renew the faithful with his strength.”
Thus we have an occurrence of the verb with ruah as subject that cannot refer to a wind blowing or sweeping and quite naturally describes the Spirit of God “hovering upon,” or as we might say in English “abiding with,” the poor.
Deuteronomy 32 revisited
You may have noticed I started my quote of Deuteronomy at 32:9 even though the context wasn't necessary to determine the meaning of verb. In 32:10, the word translated waste is the Hebrew noun tohu. While not quite as rare as rachaph, this is another rare word. And it also occurs in Genesis 1:2 where it is translated “without form.” The proximity of two rare words and the overall picture of the passages leads Kenneth Mathews to conclude that Deuteronomy 32:10-11
is probably a deliberate echo of Genesis 1:2.
Whether it is a deliberate act of the human author or not, I cannot say, but the connection definitely seems to be there. Just as God created the world out of the formless chaos in Genesis with loving care, He also carved out a land from the worthless desert for His chosen people.
So both the association of the verb with birds and the parallels with Deuteronomy suggest the active, personal presence of God in Genesis. Adopting the translation “the [S/s]pirit of God was hovering” makes better sense of the immediate context as well.
Returning to Genesis 1:2, there are three clauses in our sentence 1) “The earth was without form and void”; 2) “and darkness was over the face of the deep”; and 3) “And the Spirit of God was hovering over the face of the waters.”
The deep in the second clause is the watery abyss, possibly conceived as bottomless, from which the land emerged. So the picture painted by the first two clauses is a dark empty world over dark endless water. The third clause is largely parallel to the second, but with a decided twist. The darkness being over the deep is contrasted with ruah elohim hovering over ordinary water. Without God, there was nothing, but God has entered the scene and creation is about to come to life and Gods speaks let there be light in verse three. An impersonal wind sent by God doe not provide the proper contrast. Furthermore, if Genesis 1:2 does speak of a wind, the wind does not seem to be doing anything, as there is not anything to act on. Thus we have three good reasons to conclude “Spirit of God” is the correct translation.
Spirit or spirit?
One final question remains – should Spirit by capitalized or not. That is, does the verse speak of the Christian conception of the Holy Spirit or merely God's presence in a generic way.
In fairness, I don't think this is a question that can be definitively answered. However, there is certainly nothing in the text that would rule out the Holy Spirit and the active presence of God “hovering” could strongly imply a person of the Trinity. Furthermore, the Spirit is sometimes depicted as a bird and the verb here has strong avian connotations. Nothing definitive, but the suggestion is valid. I am fine with saying that the original audience would not have seen the Trinity here, but due to the divine aspect of scripture the sense was there, hidden only to be revealed at the right time.
That said, there is a smoking gun that demonstrates Jewish thought recognized the possibility before the coming of Jesus. Among the Dead Sea Scrolls is a text labeled 4Q422, or descriptively the cave 4 Genesis and Exodus Paraphrase. The text is very fragmentary, but as the name indicates it is thought to be a paraphrase of Genesis and Exodus. In the creation account, the second legible line is reconstructed to read
his work which he had done, and his holy (kodesh) spirit (ruah).
We don't know the verb the final phrase takes as the text breaks off there, but the presence of kodesh means that ruah must mean “spirit” here. The construct “he,” meaning God, “and his holy spirit” strongly suggests the author of this paraphrase not only saw God's spirit as being involved in creation, but also saw the Holy Spirit as a distinct person acting in creation. Thus we have pre-Christian evidence of the plurality of God being seen in Genesis 1.
In conclusion, Genesis 1:2 refers to the Spirit of God actively involved in creation. The Spirit acts upon the primordial chaos in preparation for the days of creation that follow. The verse is compatible with Christian understanding of the Holy Spirit being a distinct person of the Trinity and there is reason to believe the Jewish people were aware of this understanding before the arrival of Jesus. In full, my translation of Genesis 1:2 might read “Now the earth was formless and desolate, and darkness was over the surface of the watery abyss, but the Spirit of God was hovering over the surface of the waters.”
Florentino Garcı́a Martı́nez and Eibert J. C. Tigchelaar, “The Dead Sea Scrolls Study Edition (translations)” (Leiden; New York: Brill, 1997–1998)
K. A. Mathews, Genesis 1-11:26, vol. 1A, The New American Commentary (Nashville: Broadman & Holman Publishers, 1996).