There is a Rabbinic prohibition on constructing a temporary canopy (ohel arai; see e.g. Shabbath 137b) a tefach or greater with walls that are also a tefach or greater. Considering that this is often/typically the case with, e.g., the popular toy Magna-Tiles, do any Rabbinic authorities discuss the permissibility/impermissibility of allowing children to use these types of building toys on Shabbat, where there is a high likelihood of their building such a structure?
THIS JUST IN: According to Orchos Shabbos chapter 9 siff 13, one must warn their chinuch aged child not to use building blocks such as Lego or the like to build a house or anything which has an Ohel of a tefach by a tefach if they are going to be using the space inside of it. And if they make this edifice it is not allowed to dismantle it.
This is after having brought in chapter 8 siff 59 various opinions whether using Lego at all is allowed on Shabbos.
For those who don't know, this seffer was written in partnership between Rabbis Shalom Yosef Gelber and Yitzchok Mordechai Rubin. They consider themselves to be true students of Reb Elyashev and follow his path of psak. I am still standing behind my idea presented in the next section from my original answer, but be aware that some random guy on the internet thinks it's muttar and a well received, albeit usually stringent, seffer of practical halacha states it is not, so as always and especially now, CYLOR
The Shulchan Aruch in the last siff of siman 315 rules stringently with opinion of the Rishonim who hold there is binyan with regards to keilim, vessels. Therefore he says when closing a barrel by stretching a cloth across it, one should leave some of the opening exposed. The Mishna Berurah there quotes other Rishonim who say there is no issue of binyan with keilim. Therefore he rules that even though we are stringent like the Shulchan Aruch, we do not have to reprimand people who are lenient, being that their actions are validated according to some opinions. This would be a good reason to allow the children to continue building in this manner if one in fact held it is a problem of Ohel Arai/Binyan. Although concerning one's own children Chinuch would negate this argument, unless it is a situation where they would continue playing with it anyways. Than we would say Mutav lihyos shogagin (better to leave their mistake as unintentional).
Reasons leaning to be stringent include the fact that the 'walls' of this ohel were built now, much like setting up the barrel to be covered, as opposed to placing the pot back on the kira (barrel like oven) whose walls (of the kira) were in place from beforehand. Also, in many situations the space underneath the 'roof' of this Ohel is used for keeping toys so as noted in M.B. #20 in that siman siff 3, there would be no allowance here of an ohel made indirectly.
But I think there is a valid argument to say building this type of Ohel would be completely allowed.
See the long Taz at the end of the siman #11, who after discussing the opinions concerning binyan by keilim, and ruling stringently like the Shulchan Aruch, makes an astounding observation. According to this stringent opinion, what is the allowance for people to place the lid of their large pot back upon the pot on Shabbos? After noting that nobody is stringent in this regard, he explains what he thinks is the true understanding about this law. Chazzal only prohibited placing a cover upon something else when the cover is not designated for use as a cover for that item, similar to the cloth used to cover the barrel. In that case alone does placing a non-covering iten on something else as a covering appear similar to as if one just 'created a roof'. That is why Chazzal prohibited it. However, a lid which is constantly used as a lid does not look similar to building a roof.
I would say that Magna-Tiles fall under the category of being designated for use as covers, much like a pot lid.
See in Rabbi Ribiat's 39 Melachos in the chapter on Boneh note 146a who quotes Kalkeles Shabbos that designating any lid for any pot before Shabbos would suffice for the Taz. Rabbi Ribiat adds that in his opinion any lid which is commonly used as a lid is considered designated.This is because we don't need real designation here, but rather we just need the item to be something used for covering as opposed to an article of clothing or such which is used for other things which is not comparable to Ohel.
I bring this last source to prove both that this Taz is brought lihalacha, and to deflect any argument that any specific Magna-Tile was not designated for use as a 'roof'.
According to the answer at Frumtoronto.com Rabbi A Bartfeld quoting Rabbi Shlomo Miller compares it to Lego and says that it is permitted.