You'd be surprised at how versatile nature is
I would expect you could get remarkably dim. Really all nature is looking for is an energy gradient which is sufficient to stave off the effects of entropy.
In the early days of life, energy came mostly from purely chemical processes (themselves empowered by light, but over a very long period of time, so dimness is not a big issue). The initial genesis of life is probably rather independent of light levels, save for perhaps the need for enough ionizing radiation to lean through to jostle things around a bit at the molecular level.
Once life takes off, its greatest foe is not the environment, but the other lifeforms themselves, so they will naturally form a balance to match whatever the environment allows.
Now if your goal is recognizable life, you may have to keep things bright. A low energy environment is going to reward different tracts of life differently. As a general rule, you can probably use the 10% rule: in a food chain, you need 10 masses of foodstock to support 1 mass of predator that eats that foodstock, but the methods used by the predator to prey will shift. We won't see as much attention on movement, because movement costs calories, and they will become increasingly valuable. Rather, there will be more focus on lying in wait, preparing for prey to arrive, and then attacking them slowly (after all, the prey will be slow, themselves).
If you want modern Earth style photosynthesis using chlorophyll, your lower bound on how dim it could get is really based on the energy required to produce chlorophyll versus the energy it can harness before breaking down. I don't think there's a well known biology formula for that, because chlorophyll breakdown will certainly depend on the particulars of your solar spectrum. More UV would probably cause the molecules to break down sooner.