Are rogue planets ever born in isolation? I'm using the term "rogue planet" here loosely to mean anything of planet-level mass that is not orbiting around a star. Popular discussions of these objects always describe them as forming around a star and then being ejected by gravitational interaction with other planets. But I would have thought that, in principle, they could form in interstellar space just by the collapse of a sufficiently small cloud of dust and gas.

I realize that a small cloud will have less self-gravity, and would therefore take more time to collapse [see the answers for a correction to this assumption] and have a greater chance of being disrupted by other forces. But has a theoretical analysis been done of the minimum mass object that could be expected to form in isolation over the time frame of the current lifetime of the universe? It would be a surprising coincidence of that minimum threshold were the same mass as the mass required for hydrogen fusion.
The minimum mass of a "planet" forming from a gas cloud (definitions of what a planet is are rather slippery, and some would say this is not a planet at all) is not determined by the time available. The collapse process is rapid - less than a million years. There is a minimum mass though, and what you are referring to is something known as the fragmentation limit.

A cloud becomes unstable and collapses if its mass exceeds the Jeans mass. The Jeans mass depends on temperature to the power of 3/2 and inversely on the square root of the cloud density.
$$ M_J \propto \frac{T^{3/2}}{\rho^{1/2}}$$
When a cloud of gas collapses, its density increases. If it is able to radiate away heat efficiently, then its temperature can remain more-or-less constant and so the Jeans mass decreases. This allows the cloud to fragment into smaller pieces.

The minimum mass that can collapse in isolation will therefore be set by the smallest value that the Jeans mass can attain as the collapse proceeds. This fragmentation limit is in turn set by the cloud becoming opaque to its own radiation, which occurs when the density becomes large enough. At this point, the cloud can no longer efficiently get rid of all the heat that is generated in its interior by the work done by gravity in squashing it. The temperature rises and the Jeans mass stops decreasing. Now, the cloud may still collapse, but it won't break into smaller chunks.

The fragmentation limit is difficult to calculate with any accuracy, because it depends on the 3D turbulent dynamics of a collapsing cloud and also on whether the cloud is spinning. It is generally thought to be in the range one-to-a-few times the mass of Jupiter (e.g. Whitworth &amp; Stamatellos 2006). This is far below the minimum mass for hydrogen fusion of about 75 Jupiter masses or deuterium fusion of about 13 Jupiter masses.

Evidence that such objects may exist can be found in answers to the related questions How are rogue planets discovered? and Is there any hard evidence that rogue planets exist?