It appears that this version of the story, about how Ella Wilcox wrote Solitude, is urban legend:

A woman named Ella Wilcox once witnessed a woman sitting quietly by herself sobbing very noticeably in the middle of a train car. At first, Ella was a little bothered by the persistent weeping, but then she noticed another passenger in the car an older gentlemen who was sitting near the rear of the car. He was telling funny stories to the passengers sitting around him. Everybody smiled and chuckled along with the old man. After a while, some of the other passengers in the car started moving. They were getting up from their seats in the front, near the crying woman, and gravitating toward the back near the man telling the funny stories. Out of this experience, Ella Wilcox wrote these well-known words: “Laugh, and the world laughs with you. Weep, and you weep alone.”

When you are weeping, you may feel alone, terribly alone, but you are not. There is One who weeps with you. There is One who will one day wipe away every tear from your eyes. This One has power over life and death. This One is Jesus Christ and he has the power to call you forth from your tomb of tears and give you life once again.

The real story behind the poem is more relatable to our daily lives: 

"Solitude" is Ella Wheeler Wilcox's most famous poem. The idea for the poem came as she was traveling to Madison, Wisconsin, to attend the Governor's inaugural ball. On her way to the celebration, there was a young woman dressed in black sitting across the aisle from her. The woman was crying. Miss Wheeler sat next to her and sought to comfort her for the rest of the journey.  In the days that followed, she could not shake off the experience, as she looked at the mirror, she could see the reflection of the crying woman. Ella set to write the opening lines for “Solitude” as she considered the image of the crying woman.