Thomas is remembered for a single episode and for an adjective the episode does not use.
In John, and only in John, Thomas is absent when the risen Jesus appears to the other disciples, refuses their testimony, and says he will not believe unless he puts his finger into the nail wounds and his hand into the side. A week later Jesus appears again and invites him to do exactly that. The text does not say whether he did. What it records is his answer, my Lord and my God, which is the highest christological confession any character makes in that Gospel and the point the scene was written to reach.
The phrase doubting Thomas is not in the passage. The Greek has Jesus tell him to stop being unbelieving and become believing, and the term rendered doubt in most English versions is a translator's choice. Reading the episode as a rebuke of scepticism is a later Christian usage; reading it as the climax of John's argument about who Jesus is fits the text better.
Thomas has two other appearances in John, and both cut against the caricature. When Jesus proposes returning to Judaea where the authorities have tried to stone him, it is Thomas who says, let us also go, that we may die with him. And it is Thomas who interrupts the farewell discourse to say that they do not know where Jesus is going, which prompts the answer about the way, the truth and the life. In the synoptic Gospels he is a name in a list and nothing more.
The name is not a name. Toma is Aramaic for twin, and John glosses it with the Greek Didymos, which also means twin. Whose twin, the canonical texts do not say.