What is the origination of the mind-stream continuum? What are the basic elements that make up the mind-stream continuum?
Gautama never described an origin. Whether existence had a beginning or end was never defined. Some traditions have taken to saying "beginningless." So if mindstreams have a beginning, the Buddha never said so. The asker was either told that the answer was irrelevant to liberation (hence not worth answering) or that the question did not apply.
The Pali canon relies on dependent origination for explaining where minds come from, but does not describe mindstreams themselves. Later commentators, such as Asanga and and Vasubandhu of the Yogacara school of thought would expound on additional layers of conciousness. From my experience, most talk of mindstreams take their source as the alaya-vijnana; itself a term that has been mentioned by some contemporary Theravadin monks (Ajahn Geoff and Ajahn Sujato).
I believe the orthodox Theravada response would be the parable of the poison arrow, and Mahayana responses would depend on the proclivities of the instructor. A Yogacarin would describe the features of the alaya-vijnana and a Madhyamika (in a Tibetan tradition) would conclude that mindstreams simply are (in conventional terms at least; illusory "like a city of gandharvas" in an ultimate sense).
There is no set "Buddhist" answer here, unfortunately.