I finally found the beginning of a solution. 
The MySQL image takes an environment variable i.e. MYSQL_DATABASE that initialize the container with the name of the database on image startup See here for full documentation.
Or read the excerpt below:
MYSQL_DATABASE
This variable is optional and allows you to specify the name of a
  database to be created on image startup. If a user/password was
  supplied (see below) then that user will be granted superuser access
  (corresponding to GRANT ALL) to this database.
Here is what I came up with:
mysql:
  image: mysql:5.6.26
  environment:
   - MYSQL_ROOT_PASSWORD=root
   - MYSQL_DATABASE=bignibou
  ports:
    - "3306:3306"
I now need a way to specify the default collation but that is another story...
edit: For those interested in specifying a different collation from the default, here are the instructions to use another config file that will override the default one. See below:
Using a custom MySQL configuration file The MySQL startup
  configuration is specified in the file /etc/mysql/my.cnf, and that
  file in turn includes any files found in the /etc/mysql/conf.d
  directory that end with .cnf. Settings in files in this directory will
  augment and/or override settings in /etc/mysql/my.cnf. If you want to
  use a customized MySQL configuration, you can create your alternative
  configuration file in a directory on the host machine and then mount
  that directory location as /etc/mysql/conf.d inside the mysql
  container.
If /my/custom/config-file.cnf is the path and name of your custom
  configuration file, you can start your mysql container like this (note
  that only the directory path of the custom config file is used in this
  command):
$ docker run --name some-mysql -v /my/custom:/etc/mysql/conf.d -e
  MYSQL_ROOT_PASSWORD=my-secret-pw -d mysql:tag This will start a new
  container some-mysql where the MySQL instance uses the combined
  startup settings from /etc/mysql/my.cnf and
  /etc/mysql/conf.d/config-file.cnf, with settings from the latter
  taking precedence.