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Suppose a classical system at temperature $T$ with one variable $m$ and a free energy $F(m)$ having a metastable and a stable minimum. Suppose the system is in the metastable equilibrium at $t=0$. My question is: what is the lifetime of the system in that state?

In other words, what is the expected time before the system moves to the stable minimum? For context, this comes from the David Tong notes of statistical field theory, when describing first order phase transitions in a magnet with an external magnetic field.

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One has to distinguish thermodynamic equilibrium and how the system evolves towards this equilibrium. The power of the statistical mechanics is that it allows describing equilibrium, without considering the interactions and mechanisms (aka residual interactions) which are necessary for the system to reach equilibrium, but not essential for its properties. E.g., if an ideal gas starts in a state that is not a Boltzmann equilibrium distribution, it will never relax to equilibrium - unless we make it non-ideal by including the interactions. However, for an equilibrium system ideal gas serves as a very good approximation.

The above is to say that knowing the potential of the system (or the Landau free energy likely implied in the OP) does not tell us much about the system arrives to equilibrium - the process can be very slow, but we always assume that we have waited for long enough. One needs a more complete model to study the relaxation dynamics. @Quillo has correctly pointed in the comments that Kramers escape is one generic approach to such system (applicable not to all, but to many of the systems) - see for a review Reaction-rate theory: fifty years after Kramers

There is of course a caveat that some systems take so long to reach the equilibrium, that they are never observed in an equilibrium state (e.g., the relaxation time is longer than human lifetime or the lifetime of the Universe.) This is when we talk about (second order) phase transitions - e.g., for a ferromagnet in thermodynamic equilibrium all the directions of the magnetization are equiprobable, but this is not what we observe in practice. We then often describe this in terms of thermodynamic equilibria corresponding to different energy minima, whereas merging of these different equilibrium states into one is a phase transition (vanishing of the ordered phase; note also that Landau function is not really free energy, which has only one minimum, corresponding to all polarizations being equiprobable.)

Having said that, the reasoning discussed earlier in this answer still applies: the equilibrium theory of phase transitions doesn't tell us anything about how the transitions happen in practice, e.g., when we force the magnetization change by an external magnetic field, or when we boil water. This requires extra information, about the dynamic mechanism of such a change.

Remark
Another telling example where the mechnanism for relaxation is manifestly absent is Ising model.

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