Why does silver (I) have a larger lattice enthalpy and hydration enthalpy than sodium?
The electron configuration of sodium ion is that of the inert gas neon (1s^2 2s^2 2p^6), a spherical shell with a positive charge at the nucleus. There are 10 electrons inside this shell.

The electron configuration of silver ion is that of krypton plus 4d^10, a spherical shell with a positive charge at the center. There are 46 electrons inside this shell.

While the slightly smaller sodium ion might have slightly more polarizing power than silver ion, silver ion has far more polarizability than sodium ion. Polarizability is the distortion of the electron cloud as a result of a point charge (or dipole). Dispersion forces are a form of polarizability where two polarizable entities polarize each other. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Intermolecular_Forces/Specific_Interactions/Polarizability

The effects of dispersion forces due to the polarizability of the anion can be seen by comparing the enthalpies. The crystal structures for the fluorides, chlorides and bromides and NaI are FCC, rock salt. AgI is wurtzite. The difference in enthalpies for the fluorides is 43, but as the polarizability of the chloride anion increases, the attraction of dispersion forces increases the difference to 125, and for bromide, 148. Even tho the crystal structure of AgI is different, it still seems to have a polarizablity-polarizability effect (difference = 181). 

This is not what you would call a predictable effect, but does a good job of explaining the trend. It is not a first-order attraction, nor a second-order polarizability, but a third-order polarizability-induced polarization effect. 

And very perspicacious of you to have observed this trend.