If mitochondria originated as bacteria outside of our cells, how did they come to be part of our reproductive cycle?
To me, it’s a bit like if a snail dug it’s way into a rabbit and then managed to be born as a baby snail inside the rabbit’s offspring, or something? 
Did the genetic coding get mixed together, or have I spectacularly missed the point?
They have their own DNA and reproduce on their own (although they cannot survive for long outside a cell). You get your first ones from the egg of your mother.
We also carry some mitochondrial genes in our nuclear DNA. This relates to genetic transmission of mitochondrial diseases.
http://www.mitocanada.org/about-mitochondrial-disease/how-is-mitochondrial-disease-inherited-2/
I suppose this can also explain why the mitochondria of sperm are decidedly weird and different from most other mitochondria.
If you instead imagine a parasite that invaded all of the tissues of a rabbit, and also infected baby rabbits in the womb so that all pups of an infected mother are also infected, it seems more reasonable.
Ah, I see, so they reproduce outside of our own reproductive cycle? Is their distribution consistent throughout our cells? How is this process managed?
They must surely be considered an independent species if they are not reproducing as part of our own dna?
The symbiosis is extremely tight. Really, just like the Borg, we have assimilated them. AFAIK, there is no wild-type mitochondrian extant in the world.
They were incorporated in some primitive (bacterial?) cell ages ago, which ultimately gave rise to ALL eukaryotic species alive today (from paramecia & fungi to trees & whales).
Some theories state that the mitochondria provided the necessary energy to permit development of eukaryotes. (I heard this on Radioloab podcast-- Cellmates )
Correcting my earlier statement, it may be more correct to say that many bacteria may be kind of like mitochondria (except that mitochondria are supremely specialized to oxidatively metabolize, and not do much else).
This wiki article explains the evolutionary theory and describes the evidence:
The history of the theory is interesting, because it’s biology’s equivalent to plate techtonics in geology. It was dismissed as a fringe theory for a long time, then the evidence piled up and it was finally accepted. Lynn Margulis was the main proponent of the theory. (Unfortunately, she went a little off the deep end late in life, with bizarre theories about AIDS and several other things.)
Re the OP question they are not “part of our DNA”, they are part of our cells. But mitochondria are obligate endosymbionts because most of their original genes have been lost or transferred to the nuclear DNA. They cannot survive or reproduce outside of the host cell.
Mitochondria do, however, still have their own small genome. Unlike nuclear chromosomes, it’s circular, betraying their origin. An interesting feature is that all of your mitochondrial DNA comes from your mother, without recombination, allowing us to track the maternal lineage, and leading to the concept of the Mitochondrial Eve.
“Their” DNA is part of “our own” DNA, just not part of the nucleus. They are unable to live or reproduce outside our own cells. They are a part of us.
Much of our nuclear DNA was actually acquired from viruses, but this is now a part of our own genome.
That’s right. There is no ‘they’ anymore than them kidneys or them RNA.
A while ago there was an attempt to pass, in the UK parliament, an act that would allow genetic engineering in human subjects. The ‘for’ camp defended the motion saying that the alterations would be on mitochondrial DNA and not human. Unfortunately nobody saw beyond this little gimmick or at least refuted it with the obvious argument Colibri just laid out.
The motion carried through. Yikes.
They can’t live on their own and we can’t live without them. We are one.
If you’re talking about reproduction from organism to organism, some mitochondria are present in eggs and after fertilization these mitochondria reproduce and populate each and every cell of the developing fetus.
The DNA encoded in the nucleus sends signals to the mitochondria instructing them to reproduce when necessary. If you exercise, the mitochondria in your muscles will reproduce more frequently than if you don’t. Mitochondria play many key roles in cancer and aging.
upon re-reading the OP’s question, I think this might have some bearing on the issue: A retention mechanism for distribution of mitochondria during cell division in budding yeast - ScienceDirect
When a eukaryotic cell divides, it actively makes sure some of the mitochondria end up in each of the daughter cells. You can imagine, way back when, that the early eukaryotic cells simply relied on random luck to get at least one mitochondria into each daughter cell, but any that didn’t get any mitochondria tended to die so there was strong selection to ensure a more active mechanism.
Just because two organisms are mutually dependent doesn’t make them “one”.
However, the relationship of eukaryotic cells and mitochondria is very deep and ancient. Even if it started as a benign parasitism and evolved into co-beneficial mutualism, many eons ago the distinction between the organisms deeply blurred. They are more deeply domesticated and assimilated by our cells than dogs or horses or cows are by people. They really are ‘organelles’ now, whatever vestiges of their bacterial origin they maintain.
BTW, is their a similar relationship with plants and chloroplasts?
Yes, all the plastids have their own DNA.
<nitpick> Baby rabbits are kittens or kits.
Fascinating answers. Thanks to those who have contributed.
Yeah, but what are baby mitochondria called?
Mites?
Mittens!
Radiolab did a thing on this very topic. It raises the point that prokaryotes existed for 2 billion years before this development occurred, allowing the development of eukaryotic cells, organisms, etc. And ponders just how unlikely this random event was.