His name is common to those of the Vikings (of Scandinavian origin), his title in particular to the berserkers, Norse warriors who are recorded in Old Norse literature to have fought in a nearly uncontrollable, trance-like fury, a characteristic which later gave rise to the English word "berserk". Most berserkers wore little to no protection, save a bear skin called a "Bear-sark", which is where the word originated from. The berserkers believed that by wearing the bear's skin, they were able call upon the bear's ferocious rage and strength in battle.
Note that real Vikings did not have horns in their helms, that is just a modern depiction.
In the movie Clerks, the character Silent Bob has a cousin in a heavy metal band that goes by the stage name "Olaf the Berserker".
Olaf's quote "Urge to kill, rising!" is a reference to one of Homer Simpson's famous lines, from the Treehouse of Horror V episode.
Brolaf's quote, "Bromacia!" is a reference to Demacians and their tendency to shout "Demacia."
Brolaf's taunt "I came here to kick butt and drink Graggy! Looks like I'm all out of Graggy" is a reference to a quote in the 1988 film They Live; the original quote was "I have come here to chew bubblegum and kick ass...and I'm all out of bubblegum."
Brolaf's other taunt "You can't arrest me, my dad owns a dealership!" is a reference to Aqua Teen Hunger Force, in which two alien bros crash land on earth, and one is constantly telling others about how his dad owns this dealership.
Brolaf's quote "Hey dude, I'm kind of a big deal" may be a reference to Ron Burgundy's quote in the movie Anchorman.
Brolaf's quote "Beatdown City, population - you" is a reference to Deadpool's popular quote "Welcome to loser land, population, YOU!"
Glacial Olaf is a reference to the various ice giants in Norse Mythology.
In Chinese splash art for Olaf's skins, one of his axes is always surrounded by lightning. This is a direct reference to Thor, the norse god of thunder and his hammer, Mjolnir.
The Brolaf skin is voiced by Riot Lomar.
Brolaf's skin art is based off of the internet image meme "Beerbarian."
The name "Sivir" is written ornately on the back of Olaf's jeans, hinting at a clothing line by the Battle Mistress.
An icon of Teemo's head can be seen on Olaf's shirt.
Brolaf's drink of choice, Graggy Ice is a reference to real-life beer Natural Ice (more commonly called natty ice, or just natty) It is cheap and commonly drunk by stereotypical poor college students. It is also inferred that Graggy Ice is made by Gragas.