al-Namrood and the people of the ditch

Al-Namrood was the king of Babylon, and he was a kaafir king. There happened between him and Ibraaheem (peace be upon him) that which Allaah describes in the verse (interpretation of the meaning): 

“Have you not looked at him who disputed with Ibraaheem (Abraham) about his Lord (Allaah), because Allaah had given him the kingdom? When Ibraaheem (Abraham) said (to him): ‘My Lord (Allaah) is He Who gives life and causes death.’ He said, ‘I give life and cause death.’ Ibraaheem (Abraham) said, ‘Verily, Allaah brings the sun from the east; then bring it you from the west.’ So the disbeliever was utterly defeated. And Allaah guides not the people, who are Zaalimoon (wrongdoers)”

[al-Baqarah 2:258]

There is no story about him and any child that can be proven, as far as we know. 

But if you are asking about the story of the believing boy and the kaafir king, and how the boy was killed and the people believed in Allaah, and they bore persecution for that, this is the story of the people of the ditch, which was narrated by Muslim (3005) from the hadeeth of Suhayb (may Allaah be pleased with him), which says that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: 

The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Among those who came before you there lived a king who had a (court) sorcerer. As he (the sorcerer) grew old, he said to the king: ‘I have grown old, so send some young boy to me so that I may teach him magic.’ He (the king) sent a boy to him so that he might train him (in magic). On his way to the sorcerer he (the boy) met a monk and sat with him. He (the boy) listened to his (the monk’s) talk and was impressed by it. So whenever he went to the sorcerer he would pass by the monk and sit with him. And when he came to the sorcerer, he (the sorcerer) beat him. He complained about it to the monk and he said to him: ‘When you feel afraid of the sorcerer, say: My family made me late. And when you feel afraid of your family, say: The sorcerer made me late.’ It so happened that there came a huge beast (of prey) and it blocked the way of the people. He (the young boy) said: I shall find out today whether the sorcerer or the monk is superior. He picked up a stone and said: O Allaah, if the affairs of the monk are dearer to You than the affairs of the sorcerer, bring death to this animal so that the people will be able to move about freely. He threw that stone at it and killed it and the people were able to pass freely. He (the boy) then came to that monk and told him (what had happened). The monk said: My son, today you are better than me. Your affairs have reached a stage where I think that you will be soon put to the test, and in case you are put to the test do not reveal my identity. That young man began to heal the blind and those suffering from leprosy and he began to cure people of (all kinds) of illness. When a companion of the king, who had become blind, heard about him, he came to him with numerous gifts and said: If you heal me, all these things here will be yours. He said: I myself do not heal anyone; it is Allaah Who heals. If you believe in Allaah, I shall also pray to Allaah to heal you. He believed in Allaah and Allaah healed him. He came to the king and sat by his side as he used to sit before. The king said to him: Who restored your eyesight? He said: My Lord. He asked, Do you have any Lord other than me? He said: My Lord and your Lord is Allaah; so he (the king) took hold of him and tortured him until he revealed the identity of the boy. The boy was then summoned and the king said to him: O boy, it has been conveyed to me that you have become so proficient in your magic that you heal the blind and those suffering from leprosy and you do such and such things. Thereupon he said: I do not heal anyone; it is Allah Who heals. He (the king) took hold of him and began to torture him. So he revealed the identity of the monk. The monk was thus summoned and it was said to him: Turn back from your religion. But he refused to do so. He (ordered) a saw to be brought and he (the king) placed it in the middle of his head and cut it until it fell apart. Then the courtier of the king was brought and it was said to him: Turn back from your religion, but he refused to do so. So the saw was placed in the middle of his head and it was cut until it fell apart. Then the boy was brought and it was said to him: Turn back from your religion. He refused to do so and he was handed over to a group of his courtiers. He said to them: Take him to such and such a mountain; make him climb that mountain and when you reach its top (ask him to renounce his faith) but if he refuses to do so, then throw him (down the mountain). So they took him and made him climb the mountain and he said: O Allaah, save me from them however You will. The mountain began to shake and they all fell down, and the boy came walking back to the king. The king said to him: What has happened to your companions? He said: Allaah has saved me from them. He again handed him to some of his courtiers and said: Take him and carry him away in a small boat and when you reach the middle of the ocean (ask him to renounce) his religion, but if he does not renounce his religion throw him (into the water). So they took him and he said: O Allaah, save me from them and what they want to do. It was not long before the boat overturned and they were drowned but he came walking back to the king. The king said to him: What has happened to your companions? He said: Allaah has saved me from them. He also said to the king: You cannot kill me until you do what I ask you to do. And he said: What is that? He said: You should gather people in a plain and crucify me on the trunk (of a tree). Then take an arrow from my quiver and say: In the name of Allaah, the Lord of the boy; then shoot an arrow and if you do that then you would be able to kill me. So he (the king) called the people to an open plain and crucified him (the boy) on the trunk of a tree. Then he took an arrow from his quiver, placed the arrow in the bow and then said: In the name of Allaah, the Lord of the boy; he then shot an arrow and it hit his temple. He (the boy) placed his hands upon the temple where the arrow had hit him and he died. The people said: We believe in the Lord of this boy, we believe in the Lord of this boy, we believe in the Lord of this boy. Some people came to the king and said to him: Do you realize that you what feared has happened? All the people now believe (in the Lord of the boy). He (the king) commanded ditches to be dug at the entry-points of the roads. When these ditches were dug, and fires was lit in them it was said (to the people): He who does not turn back from his (the boy’s) religion will be thrown in the fire or they will be told to jump in it. So they started to do that until a woman came with her child and she hesitated to jump into the fire. Then her child said to her: O mother, be patient, for you are following the Truth.” 

Your saying in your question “Hazrat Ibraaheem” is an innovated expression which is best avoided, because for the Prophets (peace be upon them) there is an expression which is used only for them, namely “al-salaatu wa’l-salaamu ‘alayhim (blessings and peace be upon them). There is scholarly consensus on this point, as was narrated by Ibn al-Qayyim in Jala’ al-Afhaam from Imam al-Nawawi (may Allaah have mercy on them both) 

And Allaah knows best.