Hudson and Bellamy, 33, met last year at a music festival and announced their engagement in April.
The singer explained the baby's name shortly after his arrival.
"For those wondering, Bingham is my mum's maiden name," Bellamy wrote on Twitter.
"Bing Russell was also Kurt [Russell's] dad. Family connections all around!" Hawn is the surname of Kate's movie-star mother, Goldie Hawn.
He added: "Mum was a warrior, Bing popped out after 4.5 hours of intense pushing!"
Bellamy says the couple decided to debut the wee fella via Twitter, in a bid to shake off paparazzi trailing them for a snap of the new arrival.
"Didn't work!" he wrote on Monday. "Wow, these paps are annoying. Thought we'd tweet pic in hope of paps backing off, didn't work!
"There should be laws protecting minors... Daily mail - expect to hear from lawyers soon."
In other bambina news...
Jessica Alba and husband Cash Warren have added to their brood - they welcomed a baby girl over the weekend.
Baby Haven Garner Warren arrived on Saturday, Alba wrote on her Facebook page.
"She was born on Saturday, weighed 7 lbs. and was 19 inches long," wrote the Spy Kids 4 star.
"Healthy and happy! Big Sister Honor couldn't be more excited about the new addition to our family. Thank you for all of your support during my pregnancy. It means the world to me."
Congrats!
Not to be outdone on the big announcement front, Hilary Duff has also revealed that she's...up the duff!
The singer and actress, 23, and hubby Mike Comrie are expecting their first child.
"This weekend, Mike and I are celebrating our one year anniversary! In memory of the special day, we decided to post some of our favorite pictures from our wedding!" she wrote on her official blog.
"I can't believe it has already been a year, time really flies when you're having fun!," Duff added. "We also want to share the exciting news that... BABY MAKES THREE!!! We are extremely happy and ready to start this new chapter of our lives."
Duff and NHL hockey star Comrie, 30, got hitched on August 14, 2010.
Whirlwind wedding
Now that's what you call a quickie!
Actress Tara Reid gets engaged and married...in one day.
After an engagement that lasted mere hours, American Pie star Reid, 35, married Zack Kehayov in Greece on Saturday.
Reid took to Twitter to share the news.
"I just got engaged!" she boomed. Hours later, she added: "Just got married in Greece. I love being a wife."
Initial reports suggested that Reid had in fact got hitched to her former beau, Danish businessman Michael Lilleun. The star later took to Twitter to clarify: "His name is Zack Kehayov."
Wedding snap here. And obligatory finger bling snap here.
Reid was previously engaged to Internet entrepreneur Michael Axtmann, but called it off in 2010.
Blogger's Briefs
This juicy lot just in...
* Hanky spanky: Lady Gaga and French fashion designer Jean Paul Gaultier need to get a room
* KISS joins Michael Jackson tribute concert line-up
* Austin Powers back for another sequel. Yeah baby, yeah?
* #definitelynotwinning: Watch Charlie Sheen get pelted, booed off stage
* Paris Hilton's alleged stalker to stand trial
*Saucer of milk: Simon Cowell won't be missed on X Factor, says Louis Walsh
* Taylor Swift suffers wardrobe malfunction onstage
* Do you Belieb? This lot certainly does. Justin Bieber fans freak out when Mexico concert sells out:
Blogger Bites Back
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