Your question raises standard Shabbat questions, which we will address briefly, along with the question about melacha (forbidden work) done by a child on Shabbat. We will not discuss the contentious question of when, if ever, it is permitted to have a child do something on Shabbat that is forbidden for an adult.[1]
The prohibition of receiving benefit from melacha done on Shabbat arises in the Talmud in two contexts. One is as a k’nas (penalty) on a Jew who violates Shabbat, in order to curtail possible benefit that results from desecrating it.[2] The second is not to benefit from melacha done by a non-Jew on Shabbat on behalf of a Jew, even though the non-Jew did nothing wrong. Regarding the latter instance, Rashi[3] says that benefiting from the melacha of a non-Jew on Shabbat is an intrinsic (rabbinic) problem of inappropriateness. Tosafot,[4] in contrast, explains it as a concern that if a Jew gets used to benefiting in such a way, he may come to ask a non-Jew in a forbidden manner to do the work for him. What about a child’s melacha?
No penalty is appropriate regarding a child, who is halachically no worse (and is likely better) than one who violates Shabbat accidentally, even if the child has reached the age of chinuch (education).[5] The question we must investigate is whether there exists a prohibition of having benefit, lest one come to ask the child to do melacha.
The gemara,[6] in discussing whether one has to prevent a minor from doing what is prohibited for an adult, relates the following story. Someone lost keys in the public domain and needed to retrieve them from that area on Shabbat. Rabbi Pedat suggested that small children be taken to the area to play so that they might find the keys and return them. Tosafot[7] wonders why, regardless of whether one can let a child take the keys, it wasn’t forbidden to benefit from the keys, as would be the case had a non-Jew retrieved them. Tosafot answers that it was permitted because the children brought the keys without having the needs of others in mind. (When a non-Jew does melacha for himself, Jews may benefit from it). The Magen Avraham[8] infers from here that if a child does melacha for someone, that person may not benefit from it on Shabbat. The Pri Megadim[9] explains that this is because of a fear that an adult will ask the minor to do melacha and he will comply.[10] One leniency that can be implied from Tosafot is that if the child acts on his own behalf but does more than his needs require, we are not concerned that he had others in mind as well, as we are when a non-Jew does the melacha.[11]
Let’s get back to your case. If your son turned the lights off and on in one act of mischief, then it was all done for his own purposes and there is no problem of receiving benefit. However, maybe he shut them and, after regretting the situation that everyone was sitting in the dark, decided later to put them back on to improve the situation for his family. In that case, there should be a problem because we look at the turning on as acting to benefit others, even if he also hoped it would save him from punishment. Without reviewing all the laws of benefit from melacha on Shabbat, let us recall one rule. If a room can be used without the melacha, even with difficulty, then the extra benefit that results from the melacha is not considered forbidden.[12] Most homes have enough light that, even if the dining room lights go off, it would still be possible to dine there. Thus, the only question was probably about reading, and that, as explained, depends on the circumstances.