A look at the potential Sherlock character and his (or her?) history in Sherlock Holmes canon and fandom...

Let's talk Sherrinford. Sherlock has been dropping clues about a third Holmes brother, Sherrinford, since Mycroft delivered this little line at the end of season three: "I’m not given to outbursts of brotherly compassion. You know what happened to the other one…" This led fans to ask the question: could there be another Holmes brother (or sister)?

The Sherrinford clues.
The Sherrinford speculation flames were further fanned in the lead up to season four when Sherlock showrunners Steven Moffat and Mark Gatiss dropped the three clue words for the upcoming episodes. The word paired with Season 4, Episode 3? Sherrinford.
In "The Six Thatchers," we got the biggest reference to Sherrinford yet when Mycroft noticed a note on his fridge with the message "13th" and made a call. He tells the person on the other end of the line: "I need to speak with Sherrinford... Yes, I'll wait." If Sherlock wanted to pique our interest, it worked. Mycroft waits for no man. (But would he wait for a girl/woman? More on that later...)

Where does Sherrinford come from?
So why does Sherlock Holmes fandom equate Sherrinford with a Holmesian brother? That's kind of a long story. Sherrinford was one of Sir Arthur Conan Doyle's potential names for the character of Sherlock Holmes. The name was found in Conan Doyle's notes.
Sherlock Holmes fandom latched onto the name and, when William S. Baring-Gould wrote a fictional biography of Holmes, he included Sherrinford as his oldest brother. As country squires in 19th century England, the eldest Holmes brother would have had to stay home to manage the family estate, freeing up Mycroft and Sherlock for different kinds of lives and adventures, so this actually makes some sense within Sherlock Holmes canon. Otherwise, it seems unlikely that Mycroft would have had the time and lack of responsibility to become a civil servant, a frequent career choice of not-eldest-brothers of the time.
It's worth noting at this point that, in Sherlock's "The Six Thatchers," Mycroft moves a Reigate Square takeaway menu to find the "13th" note. "The Adventure of the Reigate Squire" is a Sherlock Holmes story. Though it is more notable for filling in some of Watson's pre-Holmesian past, perhaps Sherlock will take a different route with its narrative introduction, using it as a chance to explain more about Sherlock's past and his connection to a potential older brother.
Other appearances of Sherrinford over the years include his turn in Doctor Who novel All-Consuming Fire, where he is the member of a cult that worships a telepathic alien slug. (Please let Sherlock draw on this interpretation.) Holmes is forced to shoot his brother in order to save Watson. In Call of Cthulhu role-playing game, Sherlock must prove his brother innocent of an accused murder. Sherrinford also appears in a Sherlock Holmes-inspired Italian comic series, where he is depicted as nine years older than Sherlock.

Sherlock likes to surprise us.
Sherlock series creators Steven Moffat and Mark Gatiss seem to find joy in tricking the show's most loyal, attentive of viewers in sometimes clever, sometimes not so clever ways. Conclusion: the Sherrinford mystery is not what it seems.
Sure, Sherrinford could be Mycroft and Sherlock's older brother, hidden away in some country estate, mental hospital, or prison cell. Or Sherlock will aim to surprise us, which means those "predictable" possibilities are out. What's left? Um, the skies the limit, but my money is on Sherrinford being Sherlock's older or twin sister ("It's never twins"), perhaps hidden away in some country estate, mental hospital, or prison cell.
Or maybe Sherrinford isn't Mycroft's brother or sister, but rather a son or daughter. Thematically, it makes sense that Sherlock would bring up a potential next generation Holmes in the same episode that saw Sherlock proudly showing Mycroft a picture of his goddaughter Rosie. It would also explain why we saw Mycroft calling Sherrinford and not anyone else. Perhaps Sherrinford is at school, which is why Mycroft had to be put through to him or her. Let's be real: If anyone is going to burden their kid with the name Sherrinford, it's going to be Mycroft.
(Even crazier theory: Sherrinford is Sherlock's daughter, "looked after" by Mycroft.)

The Redbeard connection.
Another possible answer to the Sherrinford mystery is that Sherrinford was a Holmes brother (or sister) who died. We've gotten many references to Sherlock's life as a little boy with his dog Redbeard and his fictions that he was a pirate. Could Sherrinford have been a frequent collaborator with Sherlock and Redbeard in their pirate games? Perhaps Sherrinford was Redbeard, an older brother playing the part of a pirate for his younger brother. If this is the case, Sherlock may imagine his late, older brother as a dog rather than a person to help assauge the pain of losing him.
Yeah, this is a little bit of a far-fetched theory, but that's kind of what flies when it comes to Moftiss. Besides, Moffat and Gatiss have been characteristically squirrely about confirming or denying that Redbeard even is a dog (so far, we've only seen Redbeard as a dog inside of Sherlock's head). Could this be another classic example of Sherlock trying to trick us?

Why do people keep mentioning Tom Hiddleston?
If you've spent any time speculating about Sherrinford on the internet, then you have probably come across a mention of Tom Hiddleston as the third Holmes brother. This started as a popular dream-casting by Sherlock fandom, but has grown a bit since then. Nerdist even asked Mark Gatiss about the possibility of having Hiddleston on Sherlock at last summer's San Diego Comic Con. "Who says we haven’t?" Gatiss responded.
Later at the Con, Gatiss shared a photo of himself and Sherlock star Amanda Abbington (Mary) with Hiddleston...
Yeah, that got fans all riled up. Realistically, Gatiss is probably just teasing us. Hiddleston has other projects to do. But, you never know. Sherlock is one of the most watched TV shows in Britain and America and Hiddleston has been known to do a prestige TV drama. File this one under: Probably not, but it doesn't hurt to dream.
Still, my money is on Sherrinford being a girl/woman.