Although there would be some network security questions to consider, I'm imagining the ability to configure a static website for my telephone number - not too dissimilar to setting a voicemail recording, I suppose. Let's call it webtel.
Combined with a reserved top-level DNS name to resolve on-device contact-nicknames, this would allow my acquaintances to check my website for updates by visiting http://james. - and that would resolve to an IPv6 address that is in fact my smartphone running an energy-efficient static-content webserver (so james. would resolve differently on other people's phones).
After the initial deployment of this groundbreaking technology -- and because in some ways it could be considered similar to two individuals communicating over a phone call or text message -- my telecom provider would add the ability for me to view basic statistics about my website (3kb served today), and my device might be able to report contact-related info the same way it does for telephony; '2 missed calls from basil. 1 webtel view from basil'.
Webtel: it's good to HT(ML|TP)alk.