Child who stole:
If a child stole an object, then if the stolen object is still in existence, it is required to be returned to its owner. [This applies only if it is well known that this object of the child is stolen.] If the stolen object is no longer in existence, he is exempt from paying the owner for the object. This exemption remains in place even after the child reaches adulthood. [If the child sold the object and the money he received in exchange is still in existence, some Poskim rule he is obligated to return the money to the owner.]
If the child benefited from the stolen object: Some Poskim rule that even if the object is no longer in existence, if the child benefited from the object, such as he stole food and ate it, then he is obligated to pay the full amount when he is older.
Doing repentance above the letter of the law: Even if the object is no longer in existence and he did not benefit from it, it is nevertheless proper for him to accept upon himself when a gets older a certain matter for the sake of repentance and atonement, even though he had transgressed before he became of punishable age. [He should therefore pay the owner for the stolen object. He however is not required to pay for the entire worth of the object, but rather a symbolic amount. Some however write that he should pay the owner for the entire amount, however seemingly this is only a Midas Chassidus. Whatever amount the owner receives, he should be kind and merciful to forgive the person for his childhood sin.]
Is a child who stole that is below the age of Chinuch encouraged to perform a certain matter for the sake of repentance?
Some Poskim rule that the above Teshuvah is only relevant for children who damaged when they were of age to remember what they did. If however they were of very young age, and they don’t remember the incident, they are not required to do Teshuvah at all.
Is the father of a child obligated to pay for the item his child stole?
 Michaber C.M. 349/3; see also Michaber 96/3; Rambam Gneiva 1; Mishneh Baba Kama 87a “A child, getting into an altercation with him is bad, as if they damage they are exempt while if you damage them you are liable”
 Smeh 349/7
 The reason: As he does not have enough maturity of knowledge to be held accountable. [Smeh 349/6]
 Aruch Hashulchan C.M. 349/1; Piskeiy Teshuvos O.C. 343/14
 Shvus Yaakov 1/177 brought in Pischeiy Teshuvah C.M. 349/2; Based on Michaber 235/15 who requires a child to pay for a loan when he gets older if he borrowed the money for the sake of his food. Vetzaruch Iyun on Shvus Yaakov, as how a proof can be brought from the case of a loan which is a case of business, and by business we do make a child liable being it is for his benefit, as brought in Michaber 96/3; Smeh 235/43; Nesivos Hamishpat 235/29, and the same would apply in a case of borrowing for one’s food; However here that we are discussing a stolen object, what benefit does making the child pay have for the child. Vetzaruch Iyun!
 Admur 343/11; Rama 343; Mahariy 62; Bach 343; Terumos Hadeshen 2/62; Shvus Yaakov 1/177 brought in Pischeiy Teshuvah C.M. 349/2; Based on story in Shabbos 56b; Baba Kama 98b, brought in Kaf Hachaim 343/32
Other opinions: Some Poskim rule that a child is not required to repent at all for his wrongdoing even when he becomes older. [Beis Yaakov 3, brought in Shvus Yaakov ibid] Practically, we do not rule like his opinion. [Shvus Yaakov ibid; Kaf Hachaim ibid]
 The reason: The purpose of this repentance is not for the sake of removing Divine retribution, being that he is not liable for any punishment. However the forbidden act, despite not being punishable, leaves a damaging impact and spiritual stain on the soul of the child. It is for this reason that the child should do some act of repentance for the sake of cleansing his soul of any spiritual blemish. [Bach 343: The verse in Mishlei 19/2 states that “Even without knowledge, a soul is no good.” This teaches us that even a child that transgresses a sin without knowledge causes impurity to his soul, it is thus proper for him to do Teshuvah”; Likkutei Sichos 14 p. 144 footnote 13; Sichos Kodesh Balak 5730; Shulchan Hamelech 343]
Other opinions: Some Poskim rule that a child who has reached the age of understanding what is a Mitzvah and Aveira, Onas Hapeutos, is punished for his sins. Veztaruch Iyun Gadol from all the above sources that imply he is not punished. [Noda Beyehuda Tinyana Y.D. 164, brought in Pischeiy Teshuvah Y.D. 376/3]
 Shvus Yaakov 1/177, brought in Pischeiy Teshuvah ibid; Kaf Hachaim ibid; Based on Admur an Rama ibid, based on Terumos Hadeshen 2/62
 Sefer Chassidim 692, brought in Shvus Yaakov ibid and Kaf Hachaim 343/35
 Shvus Yaakov ibid
 Sefer Chassidim ibid “However if he was very young when the action was committed and he does not remember , he does not need to pay”; Beis Yaakov 3; See Piskeiy Teshuvos 343/14; Lehoros Nasan 6/102
 See Pischeiy Choshen Nezikin 5 footnote 87; Piskeiy Teshuvos 343/14