Four of the most extraordinary passages in all of the Bible are Ephesians 2:8-9, Romans 6:23, Romans 5:15-18, and John 10:4, because they all call Eternal Life a "gift":
Ephesians 2:8-9
8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:
9 Not of works, lest any man should boast.
Romans 6:23
23 For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.
Romans 5:15-18
15 But not as the offence, so also is the free gift. For if through the offence of one many be dead, much more the grace of God, and the gift by grace , which is by one man, Jesus Christ, hath abounded unto many.
16 And not as it was by one that sinned, so is the gift: for the judgment was by one to condemnation, but the free gift is of many offences unto justification.
17 For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.)
18 Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift upon all men unto justification of life.
John 4:10
10 Jesus answered and said unto her, If thou knewest the gift of God, and who it is that saith to thee, Give me to drink; thou wouldest have asked of him, and he would have given thee living water.
The passage in John 4:10-14, where the "water of life/living water" is used to refer to Salvation, is especially remarkable because of the parallel reference in Revelation chapter 22. In one of the last verses in the Bible, a final appeal is given to trust the Savior, and the freeness of the gift of Salvation is reiterated:
Revelation 22:17
17 And the Spirit and the bride say, Come. And let him that heareth say, Come. And let him that is athirst come. And whosoever will, let him take the water of life freely.
A final verse which speaks on the freeness of this gift is Romans 3:24, where we are told that we are justified freely - totally without cost to ourselves - by the Savior:
Romans 3:24
24 Being justified freely by his grace through the redemption that is in Christ Jesus:
Clearly, God wanted to hammer the point that not only is Salvation a gift, but that this gift is a true gift - being completely free. God makes it clear that is does not play deceptive word games, neither is He an "Indian giver" - one who calls something a gift only to take it back later.
Every gift given in human history has, by definition, been entirely paid for by the giver:
"Gift" - Merriam Webster Dictionary
Something voluntarily transferred by one person to another without compensation.
If the recipient had to pay even a single cent for it, it would cease to be a gift. Our Salvation was purchased by the blood of Jesus Christ on the cross, and we were ransomed from judgement and Hell:
Mark 10:45
45 For even the Son of man came not to be ministered unto, but to minister, and to give his life a ransom for many.
God's wrath fell on Jesus in our place (1 Peter 2:24: 2 Corinthians 5:21). This is why the Bible says we are bought with a price (1 Corinthians 6:20, 7:23), that price being Christ's blood (Acts 20:28). That payment for sins, which the resurrection of Christ proved was totally completed, is what allows us to be justified freely. It's free to us because of the dreadful price paid in full by Jesus Christ.
Proponents of Free Grace theology (pejoratively called "easy believism") are often accused of teaching "Cheap Grace". This is said by those who in reality cheapen it by clinging to the filthy rags of their own righteousness (Isaiah 64:6) to justify themselves instead of resting in the payment of Jesus Christ. It's not cheap, it's free, because the price was paid in its entirety by Christ. That's why He died on the cross.
Suppose I give you a gift, but as a caveat I say - "In order to accept this gift, you must listen to everything I say for the rest of your life, otherwise you can't have it."
It has ceased to be a gift and is now a contract with a payment or reward upon completion of the terms. If Salvation is a gift, then accepting it has no terms and conditions. It has no stipulations or requirements. You do not have to do a single thing for Jesus Christ in your entire life after accepting it, otherwise it has ceased to be a gift.
Salvation is not an exchange program where you dedicate your life or will to Jesus and in exchange you get Eternal Life. That's not a gift, and it's far from free. That's why God explicitly separates the "gift" from any human works:
Ephesians 2:8-9
8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:
9 Not of works, lest any man should boast.
If something that someone has to work/obey to receive can be called a "gift", then the word "gift" means nothing at all in any practical circumstance. What definition can be ascribed to it in such a situation? It's been entirely redefined, to mean, essentially, "payment". Yet, those who teach that Salvation must be obtained or secured by obedience, yet can still be called a "gift", engage in this impossible doublespeak. Such people redefine the word to mean the exact opposite of what it means.
If someone were to give you a gift, then ask for it back - for any reason whatsoever - it cannot be called a gift. It was a loan with probationary terms and conditions.
If you give someone a gift, that thing belongs to them now. Taking it back would be theft. If you told them it was a gift, and it actually was a loan, that would make you a liar. God is not a thief, nor a liar, nor does He play deceptive word games.
This is why, in addition to calling Eternal Life a gift, God is explicit in stating that He never rescinds His gifts:
Romans 11:29
29 For the gifts and calling of God are without repentance.
The truth above could also be deduced from simply looking at what the gift is. The gift is called "Eternal Life" (Romans 6:23). By definition of "eternal", this is not something that can be lost. Eternal Life is also spoken of as a present possession many times in Scripture (John 3:36, 5:24, 6:47: 1 John 5:13), always predicated on believing, proving this is not something we have to wait until we die to receive.
More Biblical proofs that God's gift of Eternal Life cannot possibly be rescinded are covered in this article, so I won't retread everything again here.
The only reason for rejecting the gift of God in favor of one's own righteousness is simply that person being too proud to accept the gift outright.
That sin of pride is the reason why, despite God offering an eternity in bliss as a free gift, mankind insists on nullifying grace (Galatians 5:1-5) by adding their own imperfect attempts at being holy to belief in Jesus in order to justify them before God.