Is it being saved by Grace that is the gift of God, or is it faith that is the gift of God?
Great question! According to the quoted passage, faith is the gift. Salvation is also a gift, but faith must come first - as the only pathway to salvation. Faith is not a natural part of our fallen nature, so it must be "gifted" to us by God - for there is no good thing within us...
Its an interesting question. The best analyses I’ve seen suggest it is the whole chain, (grace & faith resulting in salvation) = gift of God. This is based on the gender matching issues in the terms. Paul apparently used this technique as a kind of summation device.
‘Is it being saved by Grace that is the gift of God, or is it faith that is the gift of God?’
In Greek, the text is clearer. ‘That’ is neutral in gender, whereas ‘faith’ is feminine. There is a feminine form of ‘that’ which would have been used if ‘faith’ was the subject of the relative pronoun ‘that’.
In context, it is clear that salvation is the gift. Below is a link that explains, via the larger context coupled with the specifics of Greek grammar, exactly why ‘salvation’, not ‘faith’ is the ‘gift’. I hope you find the link helpful.
http://www.apologeticspress.org/apcontent.aspx?category=11&article=1246