Mother's Day was meant to honor one mother, and the apostrophe says so.
Anna Jarvis, of West Virginia, began her campaign after her mother, Ann Reeves Jarvis, died in 1905. Her mother had earlier organized "Mothers' Day Work Clubs" to teach local women how to care for children, and in 1868 held a "Mothers' Friendship Day" that brought together mothers and former Union and Confederate soldiers. Anna wanted something more personal, a day for children to thank their own mothers.
On May 10, 1908, she organized the first official observance at a Methodist church in Grafton, West Virginia. Wikipedia says that afternoon about 15,000 people attended a service she organized at the Wanamaker Store Auditorium in Philadelphia. She picked the second Sunday in May, close to the anniversary of her mother's death, and made the white carnation its emblem.
The name mattered to her. According to the historian Katharine Antolini's dissertation, as reported by the Grammarphobia blog, Jarvis preferred the singular possessive, "Mother's Day," to stress honoring your own mother rather than mothers in general. The plural form was in circulation too. A 1908 Senate resolution used "Mothers' Day." Her version prevailed: Grammarphobia found ten standard dictionaries that use "Mother's Day."
The phrase was not entirely new. The earliest Oxford English Dictionary citation is a New York Times item of June 3, 1874, about a "Mother's Day" that Julia Ward Howe launched in 1872. Howe's day was for mothers to campaign for peace. It was a different event with the same name.
Jarvis's holiday spread fast. West Virginia's governor proclaimed it in 1910, in 1912 she trademarked the phrase "Second Sunday in May, Mother's Day, Anna Jarvis, Founder," and in 1914 President Woodrow Wilson signed the measure that made it a national observance.
Then the holiday turned into something she hated. Florists, card makers and other merchants cashed in, and by 1920, History.com says, she had openly denounced the change. In 1922 she endorsed a boycott of florists, according to the Smithsonian. She brought countless lawsuits against groups that used the name, spent most of her own wealth on legal fees, and died in 1948 after lobbying the government to remove the holiday from the calendar.