From Homestar Runner Wiki
Li'l Brudder is a drawing of a one-legged puppy that Strong Bad uses to make everyone cry, and to crack himself up. According to Strong Bad, he can make it on his own, and he's going to one day "be a quarterback" that will "throw for 2000 yards." Whenever anyone (or at least Homestar Runner and The Cheat) sees this picture, they promptly burst into tears. He even managed to tear up the mighty Strong Bad himself, who then ran off, trying to cover it up by saying he had "a food in the oven".
According to Strong Bad, he is NOT named after Strong Sad. The character of Tendafoot, on the other hand, is based on Strong Sad.
According to the 2005 SXSW Panel featuring the The Brothers Chaps, a stand-alone Li'l Brudder cartoon was started, though originally abandoned. Footage was shown of a logo and Li'l Brudder scraping his way across the screen.
The Brothers Chaps did eventually complete the solo cartoon, calling it "The Li'l Brudder and Maybe Tendafoot Too Show!" It always brings Homestar Runner to tears. On the show, he is friends with Tendafoot, a no-legged parrot and a possum with a human nose for a face.
The reason as to why he has only one leg was revealed in The Li'l Brudder Show: his mother was a beagle, but his father was a unicycle.