So rename-item works with objects that support the PSDrive interface which the SSAS database does not support natively. The FindByName method on the Databases collection returns a database object, you can just change the Name property and then call the Update method to persist this change back to the server.
eg.
Import-Module SqlServer
$Analysis_Server = New-Object Microsoft.AnalysisServices.Server
$Analysis_Server.connect("server1")
$db = $Analysis_Server.Databases.FindByName("Snapshot-4")
$db.name = "Snapshot-xy"
$db.Update()