The title Dalai Lama has been used since the 16th century. It was originally given to the third leader of the Yellow Hat order of Tibetan Buddhists, and was retrospectively conferred on his two predecessors.
The name of the third Dalai Lama (as he then became known) was Bsod-nams-rgya-mtsho. Rgya-mtsho is Tibetan for 'ocean'; it also formed part of the name of the second Dalai Lama. The honorific title 'Dalai' was conferred on the third leader by a chief of the Mongol tribes who lived on the borders of Tibet. It's simply Mongolian for 'ocean' – a translation of the Tibetan rgya-mtsho. So part of the third Dalai Lama's name became an honorific title, as a tribute from this Mongol chieftain – "presumably suggesting", as Britannica surmises, "breadth and depth of wisdom".
'Lama' is simply a Tibetan word for a guru, or teacher. So the title 'Dalai Lama' is half Mongolian and half Tibetan: Mongolian for 'ocean', and Tibetan for 'teacher'. It's conventionally translated into English as Ocean of Wisdom. I'm more than a little dubious about the etymology; but for quizzing purposes, I'd say that the best course is to stick with that.
© Haydn Thompson 2017