The System User server role has the permission Execute Report Definition by default. That's what controls whether the button is visible or not. You could remove this permission from the System User role. To change the permissions, connect to Reporting Services from SQL Server Management Studio. Then go to Security -> System Roles. Right click on System User to display the permissions, remove the check for Execute Report Definition.
If you still wanted some users to see the option, you could create another role same as system user but add the Execute Report Definition permissions. You can do this in SSMS connecting to Reporting Services...same as above.