May be you have to give a try to odoo addon website_livechat.

For me it works perfectly.

Thank You.

I accidentally came across inside, and after looking through their site www.inside.tm, it morphs a business website into a virtual storefront so business personnel can see and help their site visitors in real-time using chat. It is really the best feature. Check it out and experience it yourself!

You should try Live2Support which is most affordable, reliable, fast and secure live chat software. Live2Support is not only available for Windows, MAC , Web Browsers but also on iPhone and Android. You can interact with website visitors and customers on the move.

I recommend Click4Assistance, they are a UK based provider, their security is one of their utmost priorities and the software is very easy to learn our staff and visitors love using it. We are on their middle package but I know they do a very cheap basic option, all you need to do to add live chat is going to their website http://www.click4assistance.co.uk and start the free trial. 

I recommended Live chat solution by LiveAdmins. They offer great products along with software and also offer live chat service agent as well in very low price.