I own an OG Droid and a Droid Bionic and both phones eventually had similar audio jack problems.
On one phone I found the issue to be a wad of dust crammed in the jack which I had to fish out with a [enter your choice of small pointy tool].
It could be a result of the internal contact points being pushed back from repeated tension. This would weaken the contact with the audio cable. In your particular case I don't think this is true however.
Other things you can try:
- Pull the audio cable in and out several times wiggling it and testing it. (Ideally you'd want to test it with something that plays audio continuously so you don't have to keep hitting play)
- Getting a different and potentially better quality audio cable to test with.
- You can also try resetting to factory default which may help if the problem is software related. (As long as you have a google account synced with your phone, all your apps will be automatically re-downloaded)
- If all else fails, take it back to the phone manufacturer or your phone service provider. They should be able to fix it for a reasonable price.
- There is the option of trying to fix it yourself but that isn't recommended if you are uncomfortable working with small/delicate electronics, not to mention voiding your warranty.