I found the 20Mb free SQL Server option. I still had a website in my subscription that was stopped. When I deleted that, no website or database was there in my subscription anymore. I then created a new website via the custom website option in the azure portal. I then could select (in the wizard) a free 20 MB SQL Server database.
I uploaded my MVC application and the Code First created my database automatically. I don't know why the option wasn't showing up before but now it works.