What is the Biblical basis for thinking that Jesus died for me specifically? I know that Jesus died for the whole world, but sometime this is phrased as Jesus died specifically for me.  What is the Biblical basis for thinking that Jesus died for me specifically?
In 1 Timothy 1:12-17, Paul - or someone writing as him - says (NRSV translation):

I am grateful to Christ Jesus our Lord, who has strengthened me, because he judged me faithful and appointed me to his service, even though I was formerly a blasphemer, a persecutor, and a man of violence. But I received mercy because I had acted ignorantly in unbelief, and the grace of our Lord overflowed for me with the faith and love that are in Christ Jesus.
The saying is sure and worthy of full acceptance, that Christ Jesus came into the world to save sinners - of whom I am the foremost. But for that very reason I received mercy, so that in me, as the foremost, Jesus Christ might display the utmost patience, making me an example to those who would come to believe in him for eternal life. To the King of the ages, immortal, invisible, the only God, be honor and glory forever and ever. Amen.

The author describes himself as being the pre-eminent sinner, although (from context) we know that he does not mean that he is actually worse than other people. Rather, his intention is to say that whenever anyone sins, he or she is condemned; and that everyone who is saved from the domination of sin is saved by Christ. The author counts himself as the chief of sinners because there is no greater sinner he can hide behind. There is no excuse for us. We can all describe ourselves in this way - but equally, we can all say &quot;Jesus came to save me&quot;. And because of this knowledge, we must be careful to remember (as you point out) that the same is true for other people - Jesus did not come to save only me.
In response to your clarification:
You would like to know whether Jesus had you, specifically, in mind during his Passion. What we know from Scripture is that Jesus &quot;gave his life as a ransom for many&quot; (Matthew 20:28), and that &quot;he died for all&quot; to &quot;reconcile us to himself&quot;, &quot;not counting our trespasses against us&quot; (2 Corinthians 5:15, 18-19). This act of reconciliation unites us with him in love; &quot;our old self was crucified with him&quot; (Romans 6:6) and now &quot;we abide in him and he in us&quot; (1 John 4:13). Further, it is written (Isaiah 53:4-6 and partially quoted in Matthew 8:17):

Surely he has borne our infirmities
and carried our diseases;
yet we accounted him stricken,
struck down by God, and afflicted.
But he was wounded for our transgressions,
crushed for our iniquities;
upon him was the punishment that made us whole,
and by his bruises we are healed.
All we like sheep have gone astray;
we have all turned to our own way,
and the Lord has laid on him
the iniquity of us all.

It is certain that every sin was part of his burden on the cross, including even those sins yet to happen. If this were not so, then our sins could not be redeemed, and we could not be united with him.
Jesus certainly suffered in his crucifixion, as the Gospels attest. We might argue about the extent to which he was conscious of each sin individually. While some would say that Jesus in his earthly life had only a human intellect, the transcendent nature of the Passion is surely a plausible candidate for an exception. In any case, he felt the pain of every sin, including yours and mine, regardless of precisely how he felt it.