Shannon Morgan, a resident of Leesburg, N.J., is a self-described atheist who wanted to proclaim her disbelief in God with a custom license plate. She says she went online to the state motor vehicle commission's website and applied for a plate reading "8THEIST."

But after she submitted her request, the site informed her that it couldn't process it because the term she wanted "is considered objectionable," according to a federal lawsuit she filed on Thursday. The suit says she then tested to see whether...