Have the attitudes of 50s prevail longer and become even more repressive, so that when teen rebellion kicks in, it will do so more violently and in many more forms.
Delay the counterculture by killing off people like Leary (dies during a drug episode), Kerouac (dies in a Hiking accident), and Ginsberg (killed by a gay lover)

so something like the hippie culture doesn't start much later.
Have the French do better in Vietnam and hold on to the country longer, assisted by Nato, have the Americans massively involved in it much later, like the late 60s. And then, have it escalate massively into other countries.
Have Technology really take off, maybe use the Great Man theory where the butterflies lead to a Child Prodigy being conceived that wasn't in OTL. He speeds up the development of ICs and is a public figure, who also warns about the apocalypticism of the times. A Cyberpunk cult forms around him.
Hippie culture starts in the early 70s, but other people in the counterculture find them too soft and start TTL version of a punk subculture. Student radicals espouse both cultures.
Have Kraftwerk become really really popular in the US, to the extent that it sparks a "German Invasion" much like 60s british invasion leading to earlier acceptance of electronica. The Krautrock scene can really take off here, as it would appeal to hippies, punks and cyberpunks alike.
Earlier Islamic Revolution in Iran. Because America is tied-down in Indochina, it is more gutsy and smuggles arms to the Soviet 'Stans. Soviets then Invade Iran.
Yeah, I realize that there are multiple PODs here, but many can be reconciled as a single cause, and nonetheless this will make and interesting TL.