Clearly the headliner fabric material will have to be replaced. I would be concerned about the headliner hard backing material would either disintegrate or get moldy, and then you would have a bigger problem and cost in trying to find a replacement. I would remove it as soon as possible and take the fabric off and let it dry in the sun. You can then have someone pour water on that section of the car while you look inside to see exactly where it is coming in.

At first glance it did appear to me as GraphiteGhost said that in picture two that seam appears to have a crack in it. The wire grommet, however, seems like the more likely problem area.