In conversational or informal speech, of the 1960s and in the 1960s may be treated as equivalent, but there is a slight difference in meaning. In refers to a defined time period, whereas of refers to the historical epoch, aspects of which may not correspond to exact dates.
To say something took place in the 1960s is the basic sense of in as
- Expressing a period of time during which an event happens or a situation remains the case [ODO]
In other words, at some point between the start of 1960 and the end of 1969, there was a "women's liberation movement" (though academics use that term somewhat differently, and call the broader movement second-wave feminism). Strictly speaking, something that took place in 1959 or 1970 would not be said to have taken place in the 1960s.
To say something happened in the 1960s is very close to saying it happened during the 1960s. If it took place regularly and continuously (or nearly so) you could say it happened throughout the 1960s; continuously but more episodically, it happened over the 1960s; and if it has subsequently ceased, perhaps through the 1960s.
To say something is of the 1960s, on the other hand, means it refers to the things people associate with the 1960s as a historical era— not the 1960s so much as The Sixties. This is the same way we might speak of the Victorian Gothic architecture of Keating Hall at Fordham University, though it was built in 1935, decades after the literal end of the Victorian Age.
Of in this case is for
Indicating an association between two entities, typically one of belonging, in which the first is the head of the phrase and the second is something associated with it [ODO]
Many people associate the women's movement with other social movements of the 1960s, but that movement extended beyond the boundaries of the decade. For example, the movement arguably hit its peak in the U.S. with the passage of the (never-ratified) Equal Rights Amendment, clearly a product of the activism of the preceding years, but not a reality until 1972. Similarly, Wikipedia entitles its article Counterculture of the 1960s, because salient aspects of the phenomenon continued or were manifested long after the calendar 1960s ended.
I frequently discuss with friends the disconnection between our ("Gen-X" American) associations of decades with actual dates. The '80s, we mostly agree, began with Reagan's inauguration (over a year into the chronological 1980s), but we argue over whether the zeitgeist ended with the 1987 stock market crash, the fall of the Berlin Wall in 1989, or the invasion of Iraq in 1991.
The time scale, lastly, allows for some adjustment: Christmas is of December even though the religious season is in both December and January; Marx lived in the 19th century but the application of his ideology as an international revolutionary movement is of the 20th century.