The word 'okara', which literally means refuse, consists of the honorific 'o' and the word 'kara' (which is a paronym of 'gara,' as in chagara, meaning tea dregs), and this word was typically used by court ladies.
「おから」は絞りかすの意で茶殻の「がら」などと同源の「から」に丁寧語の「御」をつけたもので、女房言葉のひとつ。