I think you maybe like Mibew - no cost as LiveZilla, but live-time at webased-script more than. You can login anywhere to serve your clients.
Live2support's chat software gives all details of website visitors at a glance. Its web stat is Amazing. Beautiful interface, easy installation.
The best live chat software to select somewhat depends on whether you're running the 'chat server' yourself, or you're having it 'hosted'. Obviously if you're operating the chat server you'll need a server administration background (but it gives you total control of the set up). The hosted option ('in the cloud') outsources the server administration portion, so that only "chat icon" coding is required to add 'chat live now with an operator' button on any web site. Either way there's an added cost to implementing this type of service -- you just need to evaluate the cost vs reward and go from there...