How do I assign a port mapping to an existing Docker container?

You can change the port mapping by directly editing the hostconfig.json file at /var/lib/docker/containers/[hash_of_the_container]/hostconfig.json or /var/snap/docker/common/var-lib-docker/containers/[hash_of_the_container]/hostconfig.json, I believe, if You installed Docker as a snap.

You can determine the [hash_of_the_container] via the docker inspect <container_name> command and the value of the "Id" field is the hash.

1. Stop the container (docker stop <container_name>).
2. Stop docker service (per Tacsiazuma's comment)
3. Change the file.
4. Restart your docker engine (to flush/clear config caches).
5. Start the container (docker start <container_name>).

So you don't need to create an image with this approach. You can also change the restart flag here.

P.S. You may visit https://docs.docker.com/engine/admin/ to learn how to correctly restart your docker engine as per your host machine. I used sudo systemctl restart docker to restart my docker engine that is running on Ubuntu 16.04.

In Fujimoto Youichi's example test01 is a container, whereas test02 is an image.

Before doing docker run you can remove the original container and then assign the container the same name again:

$ docker stop container01
$ docker commit container01 image01
$ docker rm container01
$ docker run -d -P --name container01 image01

(Using -P to expose ports to random ports rather than manually assigning).

I'm also interested in this problem.

As @Thasmo mentioned, port forwardings can be specified ONLY with docker run (and docker create) command.
Other commands, docker start does not have -p option and docker port only displays current forwardings.

To add port forwardings, I always follow these steps,

1. stop running container

   docker stop test01
   

2. commit the container

   docker commit test01 test02
   

   NOTE: The above, test02 is a new image that I'm constructing from the test01 container.

3. re-run from the commited image

   docker run -p 8080:8080 -td test02
   

Where the first 8080 is the local port and the second 8080 is the container port.

If by "existing" you mean "running", then it's not (currently) possible to add a port mapping.

You can, however, dynamically add a new network interface with e.g. Pipework, if you need to expose a service in a running container without stopping/restarting it.