To me also, haitch is a common but ill-educated pronunciation, and really marks someone out. I'm not aware of it spreading recently; I can't say I've noticed more people saying it.

An Irish friend surprised me by telling me he'd explicitly been taught to say haitch in school.

The name appears to come from trying to say H in a language (late Latin then Old French) that no longer had an H sound: from ha to aha to ahha to akkha to akka, at which point it rhymed with vacca 'cow' and like it changed k to ch: French vache, which would be pronounced vaitch by now if we'd borrowed it into Middle English.