A few thoughts.
1. Faith is not a work, Scripture is clear on this in all areas.
2. We can nto take credit for faith, because it is not our work we are placing our faith in.
3. Paul makes a statement, We are not saved by works, but by faith. Saying faith is the gift of God (although it is considering everything we are basing our faith on) would take away from what Paul is saying, Works would not have a counterpoint.
4. Looking at sentence structure. it is by Grace (unmerited favor) we have been saved (past tense. how? by faith or works??
5. The gift.. The gift is shown my what means, it was unearned, meaning it was not a reward for work done (grace) the gift given was salvation (eternal life, adopted as sons, redeemed through the blood, justified freely etc) and the avenue by which it was received. By faith (not works)
in other words, we trusted the gift was worthy of receiving, and we needed it. thus we said yes.. if we did not trust it completely (repent) we would have said no, or tried to work to earn it..