When did "妳" become the second person singular feminine pronoun?
Short answer: Before modern times, ‘妳' already existed, but it was pronounced nǎi, and was a somewhat rare variant of 奶, which primarily means milk and by extension breasts, grandmother. Starting in the early 1900s, reformers in China used the same character as a female version of the second person pronoun 你, as it replaces the 人 (person, man) on the left side of 你 with a 女 (woman).

From Wikipedia: 

Following the iconoclastic May Fourth Movement in 1919, and to accommodate the translation of Western literature, written vernacular Chinese developed separate pronouns for gender-differentiated speech, and to address animals, deities, and inanimate objects. In the second person, they are nǐ (祢 "you, a deity"), nǐ (你 "you, a male"), and nǐ (妳 "you, a female")... these distinctions are only made in Taiwanese Mandarin; in simplified Chinese, tā (它) is the only third-person non-human form and nǐ (你) is the only second person form. The third person distinction between "he" (他) and "she" (她) remain in use in all forms of written standard Mandarin.