There are essentially two answers depending on one's religion and beliefs about God.
- Predestination;
or
- Faith and effort are required to receive and apply the gift.
In brief, we either have no power to choose for ourselves, or we have the power to choose to receive the gifts that God freely offers us. If we have power to choose, then our choices really matter, and determine our eternal outcomes. If we cannot choose, then unchangeable "fate" determines our eternal outcomes and God would be arbitrary (and evil), or impotent (and hence not God).
What do the Scriptures say?
Acts 10:34
God is no respecter of persons
1 Timothy 4:14
Neglect not the gift that is in thee, which was given thee by prophecy, with the laying on of the hands of the presbytery.
Clearly it is possible to choose to neglect gifts given to us from God, or this exhortation would be meaningless.
1 Corinthians 13:4
And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing.
Clearly all these gifts amount to nothing without charity. Therefore we cannot presume that being given one or more spiritual gifts equals the fullness of salvation and eternal life.
1 Corinthians 14:1
Follow after charity, and desire spiritual gifts
Why have an action verb if our actions do not affect the outcome? Paul clearly tells us we are responsible to seek these gifts, including charity. He also says in the preceding chapters that different gifts are given to different individuals by prophecy. If you missed out on the "faith to be saved" gift, does that mean you cannot be saved? Of course not! Paul is highlighting a path of progress--one that requires our engagement, effort and earnest desire.
For as clear as Paul is on this subject, the additional revealed Scripture of the Church of Jesus Christ of Latter-day Saints greatly clarifies this matter:
Doctrine and Covenants 88:33
For what doth it profit a man if a gift is bestowed upon him, and he receive not the gift? Behold, he rejoices not in that which is given unto him, neither rejoices in him who is the giver of the gift.
Perfectly consonant with Paul's exhortation to Timothy, a gift from God does not do us any good if we do not receive it. Check your inbox from God, open His gifts to you, and use those gifts worthily.
Doctrine and Covenants 14:7
And, if you keep my commandments and endure to the end you shall have eternal life, which gift is the greatest of all the gifts of God.
Notice the word "if". God commands all men, everywhere, to repent, have faith in Christ, and receive His authorized baptism and the gift of the Holy Ghost.
Even the gift of the Holy Ghost is not automatically received by the ordinance of the laying on of hands. Although the ordinance is required to make the gift available, individual action is required to unlock this gift.
God is not evil, nor is He arbitrary. Receiving the gifts of God is a choice we each make, every day of our lives. Therefore, what we do with His gifts and opportunities in this life really matters.
Receiving any of the gifts of God is never a passive act. It always requires faith and effort on our parts to receive them. Predestination irrespective of choice, in any salvific matter, is false.
Whether one subscribes to the doctrines revealed in the Church of Jesus Christ of Latter-day Saints or not, this dichotomy holds true: One either believes that one's choices affect the outcome , or does not believe this.
Our actions--what we do with God's gifts to us--are the best testimony of the degree of our gratitude to Him for His priceless and infinite gifts, and our actions determine whether or not we are ultimately saved by those gifts.
2 Corinthians 9:15
Thanks be unto God for his unspeakable gift.
To address your additional question,
Why would anyone in their right mind refuse such a powerful, life-changing divine gift?
No one in his right mind can refuse this gift. Yielding to temptation is consenting to be deceived. The Lord has said numerous times in Scripture that wickedness leads to darkness in the heart and mind (see Isaiah 60:2, Romans 1:21, Matthew 6:23, etc.). There never was a good reason to sin or to reject any gift from God. We must all acknowledge this in order to have a sane (sane means "well/healthy") mind ourselves.