This question asks about placing an object at rest in the International Space Station. But the ISS is a large object, large enough that gravity will vary across it and cause tidal forces that can accelerate objects away from where they were left.
How big are the tidal accelerations in the ISS? Are they big enough to noticeably move things to the ceiling and floor, and in from the ends?
Tidal forces come from gradients in gravitational fields.
Along the horizontal axis of the station, the distance to Earth stays the same, so there's no gradient and therefore no tidal forces.
To find the gradient along the vertical axis, we can simply use the derivative.
$$\left(\frac{\mu}{r^2}\right)' = -\frac{2\mu}{r^3}$$
At the the orbital radius of the ISS, this works out to a gradient of $2.6 \cdot 10^{-6} s^{-2}$
The ISS is quite flat, so you can only multiply that gradient by the handful of metres between the "floor" and "ceiling" in the modules, so in the order of $\approx10^{-5} m/s^2$
That's still about a magnitude more than the acceleration due to aerodynamic drag on the station, $\approx 10^{-6} m/s²$
It's not the dominating acceleration within the station though. Lost items end up in the air filters.
edit: thanks, uhoh, for reminding me that there's a third axis to consider, the north-south axis. Rigidity here should provide a slight effect, lower than at the vertical axis, but it isn't cancelled out by orbital velocity in the same way the prograde-retrograde axis is. Here's a NASA page mapping out all the 3 axis
Update: This is not direct answer to the question as asked (thanks to @ruakh for pointing this out in the comments), but rather a supplementary answer that outlines another gravitational factor that would affect motion of an object at rest inside ISS.
According to This NASA article, an object would move inside ISS due to the mutual gravitational force between the object and ISS.
The article suggests that due to general law of mass attraction in absence of other disturbances (such as air flow inside the station) any object inside the ISS, regardless whether it is in relative motion to ISS or without it, will eventually end up at rest touching the wall which is the closest to the common center of mass (of the object and the ISS):
The general law of mass attraction is valid even for the space station itself and causes all masses to be attracted toward the common center of mass; however due to the relative insignificance of the entire mass they are attracted at such an extremely slight acceleration that traveling only one meter takes hours. However, nonsecured objects will finally impact one of the walls of the room either as a result of this or of their other random movement, and either immediately remain on this wall or, if their velocity was sufficiently large, bounce back again and again among the walls of the room depending on the degree of elasticity, floating back and forth until their energy of movement is gradually expended and they also come to rest on one of the walls. Therefore, all objects freely suspended within the space station will land on the walls over time; more specifically, they will approach as close as possible to the common center of mass of the structure.
This phenomenon can extend over hours, sometimes over many days, and even a weak air draft would suffice to interfere with it and/or to tear objects away from the wall, where they are already at rest but only adhering very weakly, and to mix them all up.
