Last year I wrote a blog post about migrating an Azure storage account from LRS to ZRS replication.
Back then, this procedure required the involvement of Azure support via a support ticket.
Things have changed since then, and now you can initiate the migration yourself. Select the storage account you want to migrate and navigate into the Redundancy section.
You select the new redundancy type and click on Save.
A confirmation is needed, and the storage account gets submitted for conversion. The migration is started within 72 hours and can take several days for large storage accounts.
For more details, have a look at the Azure documentation.