Matthew Bellamy, Kate Hudson Deciding on Name for Newborn

Kate Hudson

Jason Merritt/Getty

updated 07/11/2011 AT 1:00 PM ET

originally published 07/11/2011 AT 2:10 PM ET

Decisions, decisions.

Kate Hudson and fiancé Matthew Bellamy are still musing over what to name their new baby boy, born Saturday night, a source tells PEOPLE.

Luckily, they have a bit more time to decide – although the actress, 32, and rocker, 33, plan to settle in London while Bellamy’s band Muse works on their next album, they’ll remain in Los Angeles while Hudson recovers from the birth.

The Something Borrowed star – who delivered her 7 lbs., 12 oz. son naturally despite having undergone a Cesarean section with Ryder Russell, 7½ – is “doing great,” says the source, adding that the couple is “beyond thrilled” with their new addition.

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