Choice C is the best answer. When describing how men and women can influence society, Beecher says the ways they can do so "should be altogether different and peculiar" (lines 11-12,"gaining ...peculiar"). In the context of the "altogether different" ways men and women can influence society, the word "peculiar" implies being unique or distinctive.
Choices A, B, and D are incorrect because in the context of the "altogether different" ways men and women can influence society, the word "peculiar" suggests something unique or distinctive, not something unusual and odd (choice A), unexpected (choice B), or rare (choice D).