Smallest nearest neighbor distance among elements
Both FCC and HCP packings have nearest neighbours as close as possible (in nature at least, the simple cube is even closer but doesn't exist). Their nearest neighbour distances in terms of $a/R$ are $2\sqrt{2}$. Here $a$ is the length of a side of the unit cell and $R$ is the radius of the atom the cell consists of. 

When you are looking for the smallest nearest neighbour distance this means that you are looking for the smallest $a$ in an FCC or HPC packing. You can translate this into: you are looking for the atom which forms an FCC or HPC packing and has the smallest value for $2\sqrt{2}R$ so for $R$.

Now, I don't know all the options but for example $\ce{Cu, Au, Ag}$ form FCC structures and $\ce{Zn}$ forms an HCP structure (see e.g. here). You can look up the atomic radii and find that Zinc has the smallest: $122$ pm and will thus have a nearest neighbour distance in the lattice of $345$ pm.