There has been a lot of discussion about the meaning behind it. Yet I don’t find it hard to understand, as it is in fact really quite simple.
 
Redbeard is a typical last name given to Pirates (In many tales there has been Captains who are named Redbeard). In ASIB, Mycroft says that Sherlock had “wanted to be a pirate” when he was a child.
 
During the mind palace sequence in HLV, we see Sherlock fantasizing about his childhood dog whose name is Redbeard. He obviously had loved that dog as you see him rushing into his arms. Sherlock love for pirates is probably why the dog got that name.
 
The point of Redbeard is that it is a piece of Sherlock childhood that brings out the sentimental side in him. Yet Sherlock has always been anything but sentimental and mushy. Which is why Sherlock got very upset when Mycroft mentioned it by saying “I’m not a child.”