While both names are distinctly unusual choices, Sir is the more uncommon of the two - with the name normally used as a title before a man's first name.
Meanwhile Rumi is a Japanese girls name, and also the last name of the famed 13th century Persian poet Jelaluddin Rumi.
Along with their boy-girl twins, the couple are also parents to five-year-old daughter Blue Ivy.
The 35-year-old singer and the New York-born rap star welcomed their twins into the world earlier this month and are both said to be overjoyed.
And Jay Z has been getting a lot of advice from his friends, including old pal Diddy.
The 47-year-old music mogul is a long-time friend of the New York-born rap star, and as the father of 10-year-old twin daughters D'Lila Star and Jessie James, Diddy has been happy to offer his help to Jay.
He shared: 'Jay-Z is the one that gives me counsel, so I was so happy to be able to give him counsel on this. And the thing that I said is, you have to be prepared for double the love.
'You are used to getting love from one child, but when you get this incredible love from two kids, and they're twins, it is truly a blessing.
'And may God bless him and Beyonce and their family and everybody out there that has kids - period. That's what really runs our lives. But to have twins is double the fun.'
But when he was asked if twins were double the work, Diddy told talk show host Seth Meyers: 'It's um, um, um ... I want to say thank you to the mother of my children.'
