The origins behind the song are sexual. Dinnah is actually a contraction of 'down there', 'the ole banjo' refers to the male sexual organs (as in Banjo string). 'In the kitchen' refers to the female genitalia. Hence over time the phrase 'somones in the kitchen going down there' has migrated to 'in the kitchen with Dinnah.'
See Herb Gusset's book 'folk lore songs and tails explained'.