Coupling strengths such as $g$ and $J$ are typically discussed in context of coupled oscillating systems, both quantum and classical. They are a measure of the rate at which excitations are exchanged between the coupled subsystems. Before explaining the meaning of $g/2\pi$ in the paper by Barends $\textit{et al.}$ (Nature 508, 500–503 (2014)), it is important to realize that similar concepts/quantities emerge in classical physics. For example, if one takes two pendulums with equal lengths coupled by a spring and sets off one pendulum while the other is initially at rest, then the two pendulums begin to exchange energy with one another. In this case, the coupling strength is really just a measure of how quickly energy moves from one pendulum to the other. In this example, if one replaces the coupling spring by a long piece of dental floss, this would correspond to the $g \rightarrow 0$ limit. On the other hand, if one replaces the springs by a rigid rod, this corresponds to the $g\rightarrow\infty$ limit.
Now to answer your question about the Barends paper, if one has two qubits that are coupled with strength $g$ (meaning that when the two qubits are made resonant with one another they repel by an angular frequency $2g$), then the time required for qubit #1 to give its excitation to qubit #2 is $$T_\text{swap} = \frac{1}{2}\frac{2\pi}{2g} = \frac{\pi}{2g} \, .$$
Since (1/frequency) = time, this explains why $g$ and $J$ are typically reported in units of angular frequency. Note that as $g\rightarrow0$ (corresponding to a capacitive coupling of 0), the time required to exchange energy between the two qubits $\rightarrow\infty$. On the other hand as $g\rightarrow\infty$ (the two qubits are coupled by a huge capacitance), the time required for a swap $\rightarrow0$.
