“Our little rosebud,” Hudson, 39, wrote on Saturday alongside the sweet snap, in which her daughter appeared to be fast asleep while snuggling inside a striped blanket. In the image, the newborn wore an adorable pink hat adorned with a simple bow.
The actress, who announced her pregnancy in February, is also mom to 7-year-old son Bingham Hawn from her relationship with ex-fiancé Matt Bellamy and 14-year-old son Ryder Russell from her marriage to ex-husband Chris Robinson.
“We have decided to name our daughter Rani (pronounced Ronnie) after her grandfather, Ron Fujikawa. Ron was the most special man who we all miss dearly. To name her after him is an honor,” Hudson explained on Instagram. Fujikawa’s father, Ron, died in October 2012.
Hudson previously said she and Fujikawa had yet to decide on a name for their daughter during a Sept. 20 appearance on The Ellen DeGeneres Show with mother Goldie Hawn, 72. “We have a couple of them and we’re going to decide when the day comes,” Hudson said.
A source recently told PEOPLE that the couple was over the moon about their baby girl and becoming parents together.
“She and Danny are great,” the source said weeks before Hudson gave birth. “Kate is bubbly with excitement every time she talks about Danny and how they are having a family together. She is crazy about Danny.”
“I was happy. I was teary-eyed because I know how badly she wanted a girl. I know Danny, I’m sure in some part of every man’s brain you’re like, ‘I’d love to have a little me.’ But when he met my brother’s little girl Rio, who’s the cutest thing in the entire world, she made him be like, ‘Okay, I want a girl,’ ” he said.