A large share of the great unsolved problems of fundamental modern physics involve issues of mass and the force that couples to mass, gravity.
Both of the most consenus hypothesized particles to extend the standard model of particle physics, the Higgs Boson and the Graviton, relate to mass. Indeed, it is quite troubling that our theory requires seperate particles to explain inertia and gravitational mass, when the equivalence of the two is one of the foundations of general relativity. Likewise, one of the most notable instances in which a standard model prediction has been proved wrong is that discovery that the neutrino is massive. Many of the more esoteric hypothesized particles to extent the standard models are used to explain "dark matter" and at least one of them has to exist for conventional dark matter theory to fit the empirical evidence.
Most of the constants that the standard model requires us to treat as "fundamental" are fundamental particle masses, and there are a lot of them, at least sixteen.
Gravity is a primary motivating force for one of the latest twists in string theory (aka M-Theory), the notion that us and just about everything but gravitons, might be confined to a brane with a lower dimensionality than the universe itself.
Gravity, as formulated in Loop Quantum Gravity, Casual Dynamic Set Theory, Causal Dynamic Triangulation, and various other approaches to explaining gravity at a quantum level with a discrete or quasi-discrete space-time geometry rather than gravitons also, of course, involve gravity.
Two of the biggest problems in astronomy involve missing mass (aka dark matter) or alternately an incorrect theory of relativity in certain weak fields, and missing energy (aka dark energy) or an incorrect theory of relativity at cosmological scales.
Then there is the issue of the Pioneer Anomoly, an unexplained deviation of our space probes from the paths they would be expected to take using standard gravitational theories.
It is also worth noting that the gravitiational constant itself is one of the least accurately known fundamental constants.
In short then, many of the remaining unsolved fundamental physics problems are intimately related. Is it too much to hope that a small number of discoveries might solve all of them?