The title refers to the 1894 American folk song "I've Been Working on the Railroad" which contains the lyric "Someone's in the kitchen with Dinah..." In that case, "Dinah" was a generic name for an enslaved African woman.
Lucy says that Dinah Shore has always been her favorite singer. Some of her favorites were "Buttons and Bows," "Blues in the Night," "Dear Hearts and Gentle People," "Pen in Hand," and "Man Around the House." Dinah asks why Lucy didn't like "Shoo Fly Pie and Apple Pan Dowdy."
Harry tells Dinah that he used to listen to her on the "Eddie Cantor Radio Show." The program aired from 1942 to 1943.
Lucy Carter reminisces about growing up in Jamestown, New York, where it snowed all the time. She says her grandfather made her a red sled. Lucille Ball grew up in Jamestown, New York, and was raised by her grandfather and her mother.
When Harry is telling Lucy about his childhood winter memories, he refers to his brother as "my brother" rather than by name. Since Lucy is Harry's sister-in-law, either Harry is referring to her late husband or a different brother. In either case, he would use the brother's name as it would be familiar to both of them.