Definiely, not basil. Basil has simple leaves unlike this plant with pinnate leaves. The leaf shape and the winged stem suggest Solanum. I am not sure it is Solanum tuberosum, though. The potato leaflets look glossier, darker, wider at the base and wrinkled. These guys are of the same opinion: http://www.hort.purdue.edu/ext/senior/vegetabl/potato3.htm
Foliage varies with cultivar. Not that I would insist this one was a potato plant. It might be another nightshade family plant. If it's a seedling that popped up it will have a constricted base near the surface, unlike a sprout from a potato tuber or fragment.
The shape and colour of the leaflets do not show any non-potato characteristics as far as I can tell, not that that would rule out any other type of Solanum. One thing to consider is how long the plant took to reach the size shown - if it grew quickly then a potato growing from a tuber is likely. Also, the relatively thick stems tend to indicate a tuber grown plant rather than a seed grown one.
lol... I failed. kinda funny I was kinda in a hurry and didnt really look at the leaves. If your son planted them then they should be something kindof regular to find.
My son is six and has planted many things in our garden like pumpkin seeds and peices of bell peppers. I do think it may be a potato plant because it is growing so fast and seems to be spreading out. No flowers yet. Maybe I'll trying digging at the roots and see if I can find anything. Is it too early? I'll keep you posted. Thanks to you all:)
Yes it's too early for harvesting potatoes, the flowers come first and then the spuds, although not all potatoes flower properly, some just grow buds that terminate before opening. The best indicator that the potatoes are ready is when the foliage dies back or starts to die back, which is typically anything between 12 and 26 weeks depending on variety. This normally happens between midsummer and autumn in temperate climates, not sure what effect the Arizona weather will have on this. Togata is right, I think you need to let that kid have his own garden space!