People talk a lot about the danger of donor conceived people who are unwittingly half-siblings getting married and/or having children together. I don't think that is the weirdest thing that could happen. Here are some story ideas I have:
1) Incest Schmincest
A woman grows up knowing she was conceived with anonymous donor sperm. As she gets older, she decides she wants to conceive her own child with anonymous donor sperm. Unwittingly chooses her biological father to be the donor (she intentionally went to a different cryo bank than her mother, but he donated all over the place) and births her own half-sibling. Only finds out the truth later when DNA testing in an attempt to find each of their fathers gets super confusing.
A child created through embryo adoption takes an autosomal DNA test and has a 100% genetic match with another person in the database. Immediately assumes he is a clone. Then wonders if his future self traveled back in time to contact him. Ultimately finds out he was part of a fertilized egg that split in two and that he has an identical twin brother old enough to be his grandfather (this would have to take place several decades in the future obviously, simply because we don't have any frozen embryos that old yet). They meet up and cause twin shenanigans in spite of the age difference. It's kind of sad though because he finds out his biological parents and multiple other siblings all lived together as a family and that most of them died before he was born. Spoiler: The kid goes all Chuck Palahniuk on the storyline and blows up the cryo bank in a confusion-fueled rage, only for his much older and -- surprise! -- dying identical twin to voluntarily take the fall. And he can because they have DNA evidence linking him to the crime scene!
3) Snowflake 2 -- The Avalanche Begins
A law passes in the US stating that, because life begins at conception, all embryos have a right to be born. Women in need of money in this dystopian future sign up to be gestational surrogates to the thousands (millions?) of cryogenically frozen embryos currently in storage, some of which were abandoned nearly a hundred years ago. The "snowflake children" are simultaneously seen as a solution to the dwindling number of newborns up for adoption, and they are doled out to waiting families, foster homes, and orphanages. Some might be sold on eBay. I'm not sure where this plot would go, but it would have to follow one of the snowflakes as an adult or it would be just too sad and awful to be endured. This one has a very Brave New World / Children of Men vibe to me.
4) The Twin Project
Scientists discover how to make fertilized eggs split, creating sets of identical twin embryos in petri dishes (this can't be manipulated yet, can it? I'm under the impression it currently just happens when it happens). The doctors at one Assisted Reproductive Technology center do this with every fertilized egg created for IVF, implanting one of the embryos in the intended mother and freezing its twin to be sold through embryo adoption. Basically a few years pass and lots of people find out they have identical twins in other families. We also find out some of the sets of twins were kept by the hospital and used for research, hence the name. If you want to get just a shade darker of a plot than this, read Never Let Me Go. It's actually about cloning, but it hit home for me in a lot of ways as a DC adult, and it's a beautifully written book.
I think #4 is the only one we don't currently have the technology to make happen (I don't think we do anyway). It's actually possible #1 has already happened in the US and no one knows yet. It may not be likely, but I think the fact that there is no record keeping preventing it is food for thought. Can a woman even request her DNA be tested against her chosen donor sperm before insemination to ensure they aren't closely related? That should come up in one of those comedy films about donors having hundreds of offspring -- the prolific donor who has kept his secret so well finds out his sister's child was conceived via anonymous sperm donor too and it's his!