Yes it's possible. If I didn't miss something, the following two steps should do it:
1. Change the "Connection Name" in the Connection Manager on the slx server: Edit the connection, click OK on the Data Link Properties window, then edit the name in the next window)
2. Select the new "SalesLogix database name" in the Data Link Manager on each computer where you open any slx application (client, admin, sync server, etc.): Edit the "Log on to" connection and choose the new SalesLogix database name (= connection name from #1) on the connection tab.
You can also make change #2 through a regedit operation. Change the value of "Initial Catalog" to the new connection name under HKEY_CURRENT_USER\Software\SalesLogix\ADOLogin\Connection1 The key Connection1 is the last used connection, Connection2 the second last used and so on.
If you are using intellisync web with slx 7.5 you need to set the Provider Alias in the connection.config file. See "Configuring Intellisync for SalesLogix" in the 7.5 implementation guide.
I'm not sure what you need to do when you are using slx web client. For sure you need to do both steps from above, then you either need to rebuild/deploy or change a config file in the wwwroot. But again, I'm not sure about that.