As Aaron and Abe build The Box™, they seem to constantly be surprised at its behavior and abilities. As they discover its abilities, they start dreaming up possible uses. Given that they are so cost-conscious, and methodical, it seems unlikely that The Box did not have an intended purpose.
What was the intended purpose of The Box?
The original purpose of The Box is to create room temperature (or at least "near room temperature") superconductivity. This is not explained directly, but it's clear from listening carefully to their conversations:
Abe: ... That's what this is, OK? That's the whole difference -- that's where the box is. See, this way we don't have to play the game where we're going back and forth coming up with new and innovative ways to make the thing colder and generally unusable in the process. See, this is coming at it from the back end, and we --
Aaron: (interrupting) And unmarketable.
Abe: (responding to Aaron) Right! And unmarketable. By coming at it from the back end, see, rather than changing the surrounding temperature, we'll just change the level that it'll conduct, the transition temperature. And, see, by bombarding these edges, I'll be able to --
Aaron: (interrupting) What he's saying is: These guys, they're dropping the ceramic -- the temperature -- lower and lower. It makes the ceramic less resistant, knocks out the interior magnetic field.
Robert: Knocks out?
Abe: Exactly.
Aaron: Exactly.
Aaron: What he's saying is --
Abe: What I'm saying is: We drop the box down on it, OK, focus our own magnetic field to the gate, knock out with the inverse what's going on inside the ceramic.
Robert: (interjecting) That's smart.
Abe: And that should change the transition temperature to something we can work with.
Aaron: What are we saying that is?
Abe: Hopefully, near room temperature.
If you are old enough, you might recall the science news around the late 1980s and early 1990s concerning breakthroughs in ceramic materials that could superconduct at the relatively warm temperatures available via liquid nitrogen. (Previously, superconductivity had been observed only in metals at liquid helium temperatures, very close to absolute zero.) There was some anticipation at the time that, since this was a whole new class of materials, it might be possible to create room temperature superconductors, but I don't think much progress has been made in the interim.
Per the conversation above (and lines immediately following), the Box is intended to produce a superconducting effect by inducing atypical magnetic states in a "type-I" superconductor material, a new approach to the problem. This would be a huge commercial benefit in and of itself, but the created effect begins to show other, unexpected properties when they build the prototype version of the Box.
Note that the Box is only a subsystem of a bigger Device they are trying to build -- the Device being the one that generates the gravity-reduction effect. The Device requires superconductive plates at its base, the type-I kind that Aaron offers to steal from work. The montage shot showing Abe and Aaron handling something with tongs and thick gloves (and telling Phillip to "grab a mask") looks very much like they are dipping the superconducting plates in liquid nitrogen (or the like). The scene showing the paper dots floating presumably means that they are testing a prototype of the Device using supercooled plates, to make sure that it works correctly before they try to install the Box subsystem.
Perhaps the more interesting question (also related to your other question) is where did they get the plans hanging on the wall, which are implied to be a blueprint of someone else's version of the Device, and which the three of them are analyzing in the discussion above?
The crew are engaged in a mixture of pure research and home-build electronics. They've posted several patents, but their best selling item is in fact a graphics card.
As part of their research effort (and presumably in an attempt to construct an even faster chipset for the aforementioned GPU), they're engaged in some experiments with low temperature helium. One of the problems they face is a persistent magnetic effect, for which they construct a box to isolate the chip from external fields.
They're dropping the ceramic, the temperature lower and lower.
It makes the ceramic less resistant and knocks out the interior magnetic field.What I'm saying is we drop the box down on it.
The interesting side-effect of The Box™ is that it somehow turns out to be a time machine, not just isolating the chip from magnetic fields but isolating whatever's inside from the passage of time.
It is a mass-reducing device. Quoting the excellent summary by qntm:
Robert's idea is to build a strange piece of hardware which can theoretically reduce the mass of an object inside it. It does this by "blocking information", cutting the object in the box off from the effects of gravity. […]
While experimenting, Aaron and Abe discover that the machine works. They put a blue weeble inside the box and register that it has decreased in mass.
