In the context of PHP+MySQL, the choice is obvious — ejabberd. Chat in the role of jabber is ready out of the box. Even under it ready browser-based clients in the spirit of JWChat is: www.ejabberd.im/jwchat-localserver
P. S. And about "cheap hared hosting" in the section "startup online business" do not tell. In our time, to take a VPS with root access for 200-300 rubles/month. Such a budget even for a student.