What dictates the lifetime of a solvated electron in a given solvent?
For those who have never had the pleasure of personally doing this, see this video. It has been known since 1807 that dissolving sodium in liquid ammonia results in a beautiful color. It was originally thought the color was due to some familiar complex instead of a solvated electron. A similar phenomenon happens with other alkali metals in ammonia.

Research in the field says that it takes, at least, forty some ammonia molecules to solvate a given electron. Metstable cavities form and their stability could depend highly on electrostatic interactions to solvate our electron. Ammonia will slowly react by evolving hydrogen gas,

$$\ce{2 NH3 + 2 e- -&gt; H2 + 2 NH2-}$$

Differences in solvent can play a huge role in the stability of our solvated electron. An analogous decomposition occurs in water,
$$\ce{2 H2O + 2 e- -&gt; H2 + 2 HO-}$$

Famously the latter occurs faster than the former, see here in comparison to my earlier link. We could say that these differences in rate reflect different stabilities of our solvated electron. Although, the addition of an appropriate catalyst to our ammonia will result in rapid evolution of hydrogen.

A more quantitative description for the difference in energies is obtained by measuring the UV-vis spectrum for a solvated electron in both water and ammonia. One will quickly notice that the band appears at higher energies in water than it does in ammonia and there are considerable differences in the band shapes (the band is wider in ammonia).

So Why Do They Differ

So here I have to make the disclaimer that no current theories quantitatively reproduce the observed phenomena, such as the absorption spectrum for a solvated electron. This question has no established/accepted answer as it is an ongoing area of research.

The self-ionization of water has an equilibrium constant on the order of $K = 10^{-14}$ and that of ammonia is on the order of $10^{-30}$. Hydronium formation occurs in the case of water and ammonium in the case of ammonia. Hydronium has a considerably lower $\mathrm{p}K_\text{a}$ than ammonium and so it’s reasonable to see why a reaction in water would be more likely with an electron. (Ammonia rarely produces a weakly acidic species, but water often produces a strongly acidic complex.)

A slightly deeper reason may be that water forms more ordered local domains/structures in solution than ammonia and this influences the rate by resulting in a larger entropy of activation when the hydrolyzed electron breaks up these larger structures, recall that $k \propto \exp (\Delta S ^\ddagger /R) $.

Just speculation though.