Naming SaaS based on existing desktop product

I’m exploring options on how to name a SaaS that’s essentially a web version of an existing desktop product with some additions. I incline to keep the brand - say, the software is named “Software” and possible names for the service are:

Software Cloud
Software Net
Software Online
Software Live

Are there any good alternatives to Cloud/Net/Online/…, or alternative prefix (instead of suffix)? Maybe it’s not good idea at all, and a new, unique name should be used, to avoid customer confusion?