Customers like to have 24/7 support because it gives them peace of mind. Many companies say they have 24/7 support but it's not ALWAYS there... But there are alot of real 24/7 support offers. It's a marketing strategy in some cases.
Alot of companies use the kayako live support but i've noticed recently that quite a few companies are having trouble with this and are finding it very buggy. Another live chat application is phplive support. This is also very popular and i've not seen too many problems with this so far. These two are paid and could cost you somewhat. A decent enough free live support application is Crafty Syntax (cslh.com) this isn't as good as the other but it is free and has everything you need and creates logs and departments etc.. Although, it's not as professional looking - But decent enough.