Saudi Arabia is far from the first place to ban names. Last year New Zealand shared a list of names with CNN that its government had refused to improve. Among those names: "Lucifer," "4Real" and "Majesty." In Sweden, the names "Ikea," "Veranda" and "Brfxxccxxmnpcccclllmmnprxvclmnckssqlbb11116" (pronounced Albin) have all been rejected under Sweden's naming law.
Sweden introduced it's name law in 1982 in a bid to stop non-nobles receiving (or giving themselves) noble names, but nowadays the law is justified as a means to prevent names that could cause offense or could cause discomfort for the person using it. It's a similar system to that used in New Zealand, as well as Germany, Denmark and Norway.
In some ways, the ban on names in Saudi Arabia seems similar: Although Basmala might sound benign to English-speakers, Arabic speakers would know it has religious connotations (it means "in the name of God"). However, the Saudi name ban comes in a time increasing repression in the country, Gregory Gause, a professor at the University of Vermont, explains, pointing to the recent announcement that the Muslim Brotherhood was a "terrorist group" and a number of other measures. "It's part of a broader crackdown on everything," said Gause, who studies Saudi Arabia.
Here's the full list of names, via the Saudi Gazette: