I was musing on something the other day, and, while I haven't fully fleshed out my ideas to my satisfaction, it keeps on sticking in the back of my mind, so here's a thread on a topic that won't die, no matter how many dead horses you beat it with - I may have messed that metaphor up.
Anyway, first off, lets define 'industrial' in a reasonable way. Obviously, coal-fired steam engines are not what we're talking about here. That said, the systematic parts of industrialization, such as the organization and specialization of labor, and the more fundamental technologies of industrialization, such as your basic labor-saving devices like, are within both the cultural and technical capabilities of the era.
The Romans did have what we would recognize as factories and there was regional specialization in various industries. They also had a relatively sizable and dispersed system of watermills that were used to a variety of purposes.
On the other hand, they also had, for much of their history, a glut of labor - particularly slave labor. I personally think that the argument that slavery is antithetical to industrialization is overhyped (industrialization was historically dominated by textiles, which were produced largely with slave labor until the ACW), but slave labor does have one key disadvantage to free labor: its inherently inefficient on a large scale. Simply put, free labor will end up where you need it, slave labor, by definition, won't (or, at least, will do so much more slowly). We also have to confront the relatively rudimentary financial system developed in Rome if we're looking for true industrialization (though now I want to start a thread on 'What if Rome had a central bank').
But lets narrow our focus on two possible changes, and ask which would be more helpful to help the Roman Empire at least semi-industrialize. Basically, which could better help get a greater economic output out of the Empire - and, in particular, but not exclusively, per capita.
Option 1: After the first major plague, the Antonine Plague, hits, a variety of labor-saving devices are developed and dispersed across the Empire. Nothing like a steam engine, like mentioned before, or even anything sophisticated enough to produce a textile mill like we would recognize was the leading edge of the Industrial Revolution. Simpler things, like spinning wheels, blast furnaces, windmills, etc. Basically, your grab bag of technical innovations that the Medieval Era had over the Romans.
Option 2: The Romans have a demographic boom. Perhaps some form of immunology is developed. Perhaps more advanced crop rotation is introduced. Or better agricultural technology, like heavier plows (a classic staple of Roman AH). Either way, the Empire is either able to support a higher population, or more resistant to disease, or both.
So, either we have more productive people, regardless of how they're organized, or more people. If you could bottle up the whole Empire in a lab (twice), which do you think would result in higher output and productivity?