You might get a better understanding of the J\phi-term by calculating the equations of motion. There you should see that in this sense, J represents a current (external source).
It might get more clear if you coupled your theory to a electromagnetic field,
\mathcal L = \frac 12 D_{\mu}\phi D^{\mu}\phi - \frac 14F_{\mu\nu}F^{\mu\nu} + J_{\mu}A^{\mu}
where D = \partial_{\mu} - ieA_{\mu} and F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}. Here I have added a source term linear in the gauge field A_{\mu} rather than \phi, since this is more familiar. Now calculate the equations of motion wrt. A_\mu:
\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0.
You should get a term \partial_{\nu}F^{\nu}_{\mu}, which represents the Maxwells equations, a term depending on \phi and a term which is just J_{\mu}. This means that the gauge field A_{\mu} can get disturbed by fluctuations of \phi (which vanishes for a charge less field e=0), and by J_{\mu}. This is nothing but an (external) source as known from electromagnetism J_{\mu} = (\rho, \mathbf J).
Similar terms added to other fields (whether it's scalar, spinor, vector...) , should be understood as generalization of this idea. More practically, a source term is rarely used as something physical in QFT. It's considered as a mathematical trick to calculate correlation functions, as haushofer mentioned.