Restore Database from Network Drive
We just had a similar problem and noted that we could use UNC-pathes to specify the source of a restore.

The restore is done by some account with rather restricted permissions and therefore it does not see the drive mappings you see.

To make it work in domain context, allowing read access to the backup file to the computer.

In other cases, when the machines aren't in the same domain giving read permission to the file to everyone might be the last resort.

Edit:

Good to know that there is a build in option to add backup devices. 
For DBAs it might be the preferred way, developers in restricted environments have sometimes to try other options depending on the rights they where given.

For the German version of SSMS. You find the tool mentioned by mKorbel at

SQL Server Management Studio -&gt; Server Objects -&gt; Sicherungsmedien -&gt; Neues Sicherungsmedium...