AIO for refusing to let my cousin rename my baby after her miscarriage?

I (27F) just had my first baby, a daughter named Lila. My cousin Emily (32F) had a miscarriage two years ago and had planned to name her baby Lila. At the time, she told the family and even posted about how she’d name her next daughter that in memory. I had no idea, my husband and I just genuinely loved the name and it felt right.

After we announced the name, Emily called me sobbing, demanding we change it. I gently explained it wasn’t intentional, but she said I was “stealing from her grief.” I apologized, but stood firm that we wouldn’t be changing our daughter’s name.

Now she’s posted a long Facebook rant about “people who commodify the names of dead children,” and half the family is cold to me. My mom says I should just change the name to “keep the peace,” but it doesn’t feel fair to me or my daughter.

Am I overreacting by refusing to rename my child?