Praise be to Allah.
Yes, children will be rewarded for doing good deeds, because of the hadeeth reported by Muslim in his Saheeh (no. 1335) from Ibn ‘Abbaas (may Allaah be pleased with them both), who said: “A woman lifted up a child and said, ‘O Messenger of Allaah, will his Hajj (pilgrimage) be counted?’ He said, ‘Yes, and you will be rewarded.’” The author of Mawaahib al-Jaleel fi Sharh Mukhtasar Shaykh al-Khaleel said, regarding the matter of children being instructed to pray when they reach the age of seven: “Al-Qaraafi said in al-Yawaaqeet fi’l-Mawaaqeet: children will earn reward for good deeds that they do because of the hadeeth (prophetic narration) of the Khath’ami woman [who lifted up her child and asked the Prophet (peace and blessings of Allaah be upon him) about his Hajj].”
Ibn Rushd said: “The bad deeds of young children are not recorded but their good deeds will be recorded, according to the sound opinion.”
Ibn ‘Abd al-Barr said in al-Tamheed, commenting on the first hadeeth quoted therein, which is the hadeeth of the Khath’ami woman: “… Abu’l-‘Aaliyah al-Riyaahi said: ‘Umar ibn al-Khattaab said: the young child’s good deeds will be recorded but his bad deeds will not be recorded.”
The author of Mawaahib al-Jaleel said, concerning children entering ihraam (sacred state) for Hajj and ‘Umrah:
“The scholars do not differ concerning the fact that children will be rewarded for whatever acts of obedience (to Allaah) they do, and will be let off for any bad deeds that they do, and any bad deeds they do deliberately will be counted as mistakes. It says in Mukhtasar al-Waadihah: ‘Hajj is not an obligation for boys and girls until boys reach puberty and girls begin menstruating, but there is nothing wrong with taking them for Hajj. It is recommended (mustahabb), and the Messenger of Allaah (peace and blessings of Allaah be upon him) did so.’”
Then he reported that Talhah ibn Musarrif said: “It was one of the customs of the Muslims to take their children for Hajj and expose them to the mercy of Allaah.”
Ibn ‘Abd al-Barr said in al-Tamheed that it is encouraged to take children for Hajj, and that the majority of scholars say this. He also said: “It is nothing strange that a child should earn reward and status in the Hereafter for his salaah (prayers), zakaah (paying poor-due), Hajj and other good deeds if he does them as they should be done, (because this is) grace and mercy from Allaah, just as Allaah shows mercy to the dead by rewarding them for acts of charity done on their behalf by the living. Do you not see that they (the scholars) are agreed that a child should be commanded to pray when he reaches the age of understanding and that the Prophet (peace and blessings of Allaah be upon him) led Anas and the orphan in prayer? Most of the salaf (pious predecessors) said that zakaah has to be paid on the property of orphans, and it is impossible that they would not be rewarded for that. Their guardians and the one who does this on their behalf will also certainly be rewarded, as will the one who takes them for Hajj, as a blessing and mercy from Allaah. It was reported that ‘Umar said: ‘Children’s good deeds will be recorded and their bad deeds will not be recorded.’ I do not know of anyone whose opinion is worth following who said anything to the contrary.”
According to al-Ikmaal, many of the scholars said: “Children will be rewarded for their acts of obedience (to Allaah), and their good deeds will be recorded, but not their bad deeds.”
It says in Awaa’il al-Muqaddimaat: “The correct opinion in my view is that they are both encouraged to do that and will both be rewarded for it (i.e., the child and his or her guardian). The Prophet (peace and blessings of Allaah be upon him) said to the woman [who asked about the child’s Hajj]: ‘… and you will be rewarded.’ And Allaah knows best.”
Ibn Jamaa’ah said: “According to the four (imams/leading scholars), children will be rewarded for their acts of obedience and their good deeds will be recorded, whether they are mature or not. This was reported from ‘Umar, may Allaah be pleased with him. Some scholars report that there was consensus (ijmaa’) in this matter. It is also indicated by what we have already said under the heading of Virtues (fadaa’il), that the Prophet (peace and blessings of Allaah be upon him) said: ‘The jihaad (physival battle) of the very old and the very young is Hajj and ‘Umrah,’ as well as the hadeeth about the woman who lifted up a child.”
And Allaah knows best.