Coriolis forces (yea, spelling, I know) from the earth's slight wobble as it rotates would cause the can to rub on the side of the hole so there would be some energy losses both dropping to the center and coasting on the way back up on the far side. Even if those forces are very small, the can would climb to just below the distance from the center of the earth's mass as it was dropped. Assuming a perfectly spherical earth with uniform density (which is not actually true), if the can was dropped from the exact entrance to the hole, it would not quite make it back up to the surface on the far side. It would cycle back and forth until those very small energy losses cancelled out all of the can's potential energy (dist from earth multiplied by mass) and the can would then be at rest in the center of mass. Of course, all that energy would be dissipated as heat, absorbed partially by the can and partially by the side of the hole (the earth). So it would increase the overall heat of the earth. If you look at the total amount of energy in a can at the surface of the earth relative to it's position at the center of the earth, that's a heck of a lot of heat so unless there was a way to shed all that heat, the can would likely be vaporized long before it quit moving.
Of course, that assumes that all of the air in the hole is evacuated so the can is falling through vacuum. Although this is impossible, it is no more impossible than the original premise of drilling a hole straight through anyhow.
A can-sized object falling to earth from orbit would vaporize from air friction fairly high up in the atmosphere, and that's just the energy being dissipated from an orbit a couple hundred miles up.
Now if the hole was filled with air pressurized to the "sea level" (again impossible), the can would reach terminal velocity somewhere around 200 miles per hour and so it would be effectively braking the entire way down. So it would climb maybe a few hundred feet on the far side of the earth's core and stabilize at the center a lot faster.
If the air increases pressure on the way down to the center of the earth, the can will crumple and turn into a little hot wad of metal, and again probably vaporize from the heat and pressure. Then again, at the pressures present at the center of the earth, I think oxygen and nitrogen turns into a liquid and acts like a metal, so that is sort of a silly condition.
So... Lots of different answers depending on your assumptions.
