While making some adjustments to the back of my system, the HDMI cable fell out of my Sony Blu-ray player. When I attempt to put the cable back into the jack, it's so loose that it won't make a connection. Is there an easy way to tighten the HDMI jack so I can reconnect via HDMI or do I need to bring it to a Sony repair center to have it fixed?

One thing you can do is to visually inspect the HDMI cable as well as the HDMI receptable in the Blu-ray player itself, and check that none of the pin have been damaged or pushed back.

Does the cable just not get fully seated anymore, or does it fully seat but has no retention force?

It doesn't get fully seated. Today, I brought the Blu-ray player in to be repaired. It's a little over 2 months old, so it's still under full warranty. Why bother attempting to fix it myself when I can get it fixed properly for free.

I have the same issue, is there a way to fasten the cable to the player itself, or would repair be necessary?

If the cable doesn't hold itself in (especially if it's horizontally mounted, without any external forces pulling on i), then something is out of spec and/or damaged. All cable connectors should have a good bit of friction when plugged into any compliant receptacle.

The obvious troubleshooting would be to try another brand of cable to see if that is tighter.