Lydia was not chosen to BELIEVE since she was a worshiper already – Acts 16:14

And a certain woman named Lydia, a seller of purple, of the city of Thyatira, WHICH WORSHIPPED GOD, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul (Acts 16:14)

Lydia was never hand-picked to be a believer. God doesn’t work in such a way that he randomly (or for unknown reasons) selects a few lucky people to give faith and salvation while he leaves the rest for damnation, despite being guilty of having prevented them from believing by not enabling them to believe. That would hardly be righteous actions worthy of the God of the Bible. We are saved by FAITH and not by ELECTION. If we refuse to believe, we should not blame God for this failure but only ourselves.

Lydia found Paul and some other christians down by the beach where they were praying. Since she was a worshiper of God already, she was probably intrigued by what she heard and stayed to listen to learn more. That was a good choice since she learned more about the kingdom of God.

13 And on the sabbath we went out of the city by a river side, where prayer was wont to be made; and we sat down, and spake unto the women which resorted thither.14 And a certain woman named Lydia, a seller of purple, of the city of Thyatira, WHICH WORSHIPPED GOD, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul.15 And when she was baptized, and her household, she besought us, saying, If ye have judged me to be faithful to the Lord, come into my house, and abide there. And she constrained us.

It’s true that it says “whose heart the Lord opened”, but we already know God didn’t open hear heart to be able to believe or to be a worshiper of God since she was this already. Her faith was the reason she was compelled to approach them and listen to their message.This was her own decision in the same way that Jesus disciples decided to either stay with him and learn more of his teaching, or to leave altogether (as in John. 6:66) and therefore remain in darkness. The reason that God opened her heart even more was so that she could “attend to the things spoken by Paul” (Young’s literal). So not “to believe” in the first place but to be able to understand Paul’s advice – perhaps about the most fruitful way to live a christian life and to be a useful tool for others. The Holy Ghost can certainly work in us, but we are the ones who must make the decision to repent and seek the Lord before this can begin to happen. We are requested to repent (and be baptized for the remission of sins, which should be made right after repentance) and then we will receive the gift of the Holy Ghost.

Luke 11:13 If ye then, being evil, know how to give good gifts unto your children: how much more shall your heavenly Father give the Holy Spirit to them that ask him?

Acts 2:38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.

Acts 17:26 And hath made of one blood all nations of men for to dwell on all the face of the earth, and hath determined the times before appointed, and the bounds of their habitation;27 That they should seek the Lord, if haply they might feel after him, and find him, though he be not far from every one of us:28 For in him we live, and move, and have our being; as certain also of your own poets have said, For we are also his offspring.29 Forasmuch then as we are the offspring of God, we ought not to think that the Godhead is like unto gold, or silver, or stone, graven by art and man’s device.30 And the times of this ignorance God winked at; but now commandeth all men every where to repent:

What else does the same chapter say about Lydia and about having faith? – Acts 16

Churches increased in number daily due to the word that Paul and Silas spread. What would have happened if they did not spread the gospel? Would the result be the same?

4 And as they went through the cities, they delivered them the decrees for to keep, that were ordained of the apostles and elders which were at Jerusalem.5 And so were the churches established in the faith, and increased in number daily.

God wants us to spread the gospel since people might be saved through it. If we are saved by election (maybe from the creation of the world), spreading the gospel doesn’t seem to be important since it won’t make a difference:

9 And a vision appeared to Paul in the night; There stood a man of Macedonia, and prayed him, saying, Come over into Macedonia, and help us.10 And after he had seen the vision, immediately we endeavoured to go into Macedonia, assuredly gathering that the Lord had called us for to preach the gospel unto them.

The Philippian jailer fell down before Paul and Silas, likely knowing they served the one true God who can perform amazing miracles – such as creating earth quakes at the right time and open prison doors. He asked them “what must I DO to be saved?“.

27 And the keeper of the prison awaking out of his sleep, and seeing the prison doors open, he drew out his sword, and would have killed himself, supposing that the prisoners had been fled.28 But Paul cried with a loud voice, saying, Do thyself no harm: for we are all here.29 Then he called for a light, and sprang in, and came trembling, and fell down before Paul and Silas,30 And brought them out, and said, Sirs, what must I DO to be saved?

Paul and Silas did not answer: “Absolutely NOTHING, since you have to hope for that God will possibly open your eyes so that you can believe. IF God works in you in such a way, then you know you’re one of the elect and you’ve been elect from birth and even from the creation of the world. You can’t possibly affect a decision about your destiny that has been made thousands of years ago. You can just relax and wait and see.”

Instead Paul and Silas answered his question with:

31 “—BELIEVE on the Lord Jesus Christ, and thou shalt be saved, and thy house.”

They did not promise automatic salvation for his entire household if only this jailer would take the path leading to salvation, because the same condition applied also to them. “BELIEVE and thou shalt be saved!” Paul and Silas might also have figured out that this jailer likely would take the step to be saved since he had just witnessed a miracle and freely requested to know what he must do to be saved – showing his interest in his heart. If the man of the household gets born again, it’s likely that he will be able to affect his entire household with his new spiritual life, and especially if Paul and Silas evangelized to them too, which they did:

32 And they spake unto him the word of the Lord, and to all that were in his house.

The jailer’s family indeed ended up BELIEVING, so they were not saved only because the man of the household was a believer. Neither did Paul and Silas contradict James (in his letter) where James explained that faith without deeds is dead. Also demons believe and tremble, so “faith” means nothing unless you show your faith with your deeds. Abraham didn’t just sit under a bush and believed without obeying God. No, he showed time and again that he had a strong faith by being obedient to God, even when it came down to offering up his own son.

33 And he took them the same hour of the night, and washed their stripes; and was baptized, he and all his, straightway.34 And when he had brought them into his house, he set meat before them, and rejoiced, believing in God with all his house.

Paul and Silas left prison and stayed in the house of Lydia for a while.

39 And they came and besought them, and brought them out, and desired them to depart out of the city.40 And they went out of the prison, and entered into the house of Lydia: and when they had seen the brethren, they comforted them, and departed.