A density functional for the lattice gas (Ising model) from fundamental measure theory is applied to the problem of droplet states in three-dimensional, finite systems. Similar to previous simulation studies, the sequence of droplets changing to cylinders and to planar slabs is found upon increasing the average density $\barρ$ in the system. Owing to the discreteness of the lattice, additional effects in the state curve for the chemical potential $μ(\barρ)$ are seen upon lowering the temperature away from the critical temperature (oscillations in $μ(\barρ)$ in the slab portion and spiky undulations in $μ(\barρ)$ in the cylinder portion as well as an undulatory behavior of the radius of the surface of tension $R_s$ in the droplet region). This behavior in the cylinder and droplet region is related to washed-out layering transitions at the surface of liquid cylinders and droplets. The analysis of the large-radius behavior of the surface tension $γ(R_s)$ gave a dominant contribution $\propto 1/R_s^2$, although the consistency of $γ(R_s)$ with the asymptotic behavior of the radius-dependent Tolman length seems to suggest a weak logarithmic contribution $\propto \ln R_s/R_s^2$ in $γ(R_s)$. The coefficient of this logarithmic term is smaller than a universal value derived with field-theoretic methods.

14 pages, 8 figures, to be published in Phys. Rev. E