James Akin has a nice explanation on why women take their husband's last name upon marriage and what was behind the push for women to keep their own names.
In patrilineal cultures when a marriage occurs the wife becomes part of her husband's family, and if you have family names in such a culture, it becomes natural for the wife to take her husband's family name.
There may be a sense in such cultures that both the husband and wife are really part of each other's families now, but since descent is reckoned by the male line, there is a greater sense that the wife is part of the husband's family rather than visa versa.
These are the reasons that the custom exists anthropologically, but the origins of the custom tended to be obscured in the minds of many.
When radical feminism came along, it wanted to radically tinker with the sexual status quo, to smash traditional gender roles, and even to call into question the institution of the nuclear family. (I'm talking about radical feminism, mind you, not moderate feminism that merely wanted better treatment for women.)
Doing away with the historical naming conventions would serve those goals (as well as making it harder to keep track of who is related to whom, thus undermining the family), and so not taking the husband's name became a symbol of defiance against traditional values.
He makes it clear that it's a question of culture rather than one of theology. Many cultures operate differently than the US, and he carefully suggests that motives for bucking the system will vary. The comments he's accrued on the subject are worth reading as well.
Comments