I take your third option as the best because we are saved, or justified “by grace through faith,” which is how we receive Jesus Christ and his obedience on our behalf. In case we do not get it, the opposing competitor for such salvation is stressed so that it is absolutely clear.
Works is not a competitor to grace, or faith, but to justification and salvation. Salvation is “not of works, lest any man should boast.” Salvation is not of works, yes this is the gospel. The gospel does not try to argue that grace or faith is not of works, but always argues that salvation is not of works. Faith produces works, so it would be confusing to say that faith is not of works without further explanation.
The ‘works’ that the Apostle is ‘excluding from our salvation’ are works from Christians. That is why he naturally transitions after this to the idea that we are his workmanship. This is to say that salvation is free and the works it produces in us have nothing to do with our salvation (justification) but are his work for his own pleasure.
The apostles basic method is always to separate works from salvation not always to create the doctrine of irresistible grace, though that doctrine can be argued in various ways from other places. I think this is the most face-value interpretation of the words and delving into the Greek does not help, as the questions pertains to ‘this’ – ‘this grace’, ‘this faith’, or ‘this salvation’ as you have succinctly shown in the question.
Once again you have forced me to make up my mind about a particular that I had not decided before – thanks again for the great question.
Here is some additional analysis (the longer version) as requested by a user:
Looking at the Greek one can easily see that all the common English translations are fine without any controversy leading to one answer or another:
For [γὰρ] by grace [χάριτί] you are [ἐστε] saved [σεσωσμένοι] through [διὰ] faith [πίστεως] and [καὶ] this [τοῦτο] not [οὐκ] of [ἐξ] yourselves [ὑμῶν] God's [θεοῦ] - [τὸ] gift [δῶρον] not [οὐκ] of [ἐξ] works [ἔργων] that [ἵνα] not [μή] anyone [τις] might boast [καυχήσηται].
Either grace χάριτί is the gift, or our being saved σεσωσμένοι is the gift, or our faith πίστεως is the gift – that is ‘not of works that no anyone might boast’. Therefore that the Greek can’t help as the common English translations are already accurate without controversy and do not indicate which word is 'the gift'.
Therefore we turn to the internal logic of scripture that transcends human language.
This is what we speak, not in words taught us by human wisdom but in words taught by the Spirit, expressing spiritual truths in spiritual words. The man without the Spirit does not accept the things that come from the Spirit of God, for they are foolishness to him, and he cannot understand them, because they are spiritually discerned. (NIV 1 Corinthians 2:13-14)
Obviously grace, faith and salvation are the kinds of words Paul refers to as ‘spiritual words’. We must therefore look for the maning of these words in humility and with faith and with the study of the Bible's overall arguments, especially those of the Apostle Paul.
The only question to answer then is 'Why using an understanding from Paul’s arguments, would one choose ‘salvation’ as the gift rather than ‘faith’ or ‘grace’?' Certainly all three can be seen as gifts, but not all three are constantly set up by the Apostle as being opposed to God in terms of how we are saved and in terms of how it derives a foolish kind of boasting. Paul always argues that 'salvation' or 'justification' is not of works, he does not say 'grace is not of works under the law', or 'faith is not of works under the law'.
'Works' is something in the New Testament as causing (primarily the Jews) a reason to ‘boast’ based on ‘works of the Law’. (Romans 2:17). The purpose of those works is to become ‘justified before God’ giving a man a reason ‘to boast’. (Romans 4:2). Now justification is nothing more than our salvation, so the question is, 'Are we justified/saved by 'works' or by 'faith' in Jesus Christ? Well typically Paul say’s:
Therefore, since we have been justified through faith, we have peace with God through our Lord Jesus Christ, through whom we have gained access by faith into this grace in which we now stand. And we boast in the hope of the glory of God. (NIV Romans 5:1-2)
So we see the idea; salvation is either a result of faith (giving us access to grace which justifies us) or by works under the law (from which we can boast). With faith our boast changes into the glory of God, and not of ourselves.
What then is 'not of our own doing'? Clearly it is our 'justification', because by works of the law the Jews tried to justify themselves and boast. Paul does not say that the Jews by works of the law sought to obtain faith. Paul does not say that the Jews by the works of the law sought to obtain grace. The Jews did not boast in grace, or faith to obtain justification but works. Clearly justification which is 'being saved' is either obtainable by 'works' or is given as a 'gift'. This the Apostle constantly puts as competing principles, either one or the other must stand or fall. The Jews were not trying to obtain grace, or faith by works. Therefore the 'gift' that is opposed to works is 'justification/salvation' and not faith or grace.
In this way we could read the original text like this:
For by grace you have been justified through faith. And this justification is not your own doing under the law; it is the gift of God by grace, not a result of works under the law, so that no one may boast in themselves. (My Paraphrase)