In real life people name their children after family members, celebrities, people they admire and book characters all the time. The next time a high profile celebrity couple has a child just look at the boost in baby names of their child's name.
Also note as with many families, at least those I know of in the UK, names are passed down through the generations. My first name is a name passed down through all known generations on my dad's side and my middle name is a name that comes from my granddad on my mum's side (it could even go further than that but I'm not sure).
With this in mind let's have a look at why Brandon is a popular name in House Stark.
Brandon the Builder
Legend has it that Brandon the Builder is the founder of House Stark, raised Winterfell, the Wall, Storm's End and a 200 foot tower in Oldtown; though the tower may have been designed by his son Brandon.
The greatest castle of the North is Winterfell, the seat of the Starks since the Dawn Age. Legend says that Brandon the Builder raised Winterfell after the generation-long winter known as the Long Night to become the stronghold of his descendants, the Kings of Winter. As Brandon the Builder is connected with an improbable number of great works (Storm's End and the Wall, to name but two prominent examples) over a span of numerous lifetimes, the tales have likely turned some ancient king, or a number of different kings of House Stark (for there have been many Brandons in the long reign of that family) into something more legendary.
The World of Ice and Fire, The North: Winterfell
That tower, we are told, rose two hundred feet above the harbor. Some say it was designed by Brandon the Builder, whilst others name his son, another Brandon; the king who demanded it, and paid for it, is remembered as Uthor of the High Tower.
The World of Ice and Fire, The Reach: Oldtown
Brandon the Builder allegedly also gave the lands south of the Wall to the Night's Watch though again this could have been another Brandon.
"The Night's Watch," he answered. "This is the Gift. The New Gift, and north of that Brandon's Gift." Maester Luwin had taught him the history. "Brandon the Builder gave all the land south of the Wall to the black brothers, to a distance of twenty-five leagues. For their . . . for their sustenance and support." He was proud that he still remembered that part. "Some maesters say it was some other Brandon, not the Builder, but it's still Brandon's Gift. Thousands of years later, Good Queen Alysanne visited the Wall on her dragon Silverwing, and she thought the Night's Watch was so brave that she had the Old King double the size of their lands, to fifty leagues. So that was the New Gift." He waved a hand. "Here. All this."
A Storm of Swords, Bran III
Brandon the Shipwright and Brandon the Burner
Both are members of House Stark the Shipwright loved the sea and tried to cross the Sunset Sea but was lost to it. After his father failed to return the Burner set light to the Stark's ships and ended their power at sea.
That's a Brandon, the tall one with the dreamy face, he was Brandon the Shipwright, because he loved the sea. His tomb is empty. He tried to sail west across the Sunset Sea and was never seen again. His son was Brandon the Burner, because he put the torch to all his father's ships in grief.
A Game of Thrones, Bran VII
Brandon Stark Son of Rickard Stark
Perhaps the most obvious but this is Eddard's brother and Bran's uncle and who perhaps Bran is named after. Especially as Brandon's life was cut short due to the Mad King.
"And there's my grandfather, Lord Rickard, who was beheaded by Mad King Aerys. His daughter Lyanna and his son Brandon are in the tombs beside him. Not me, another Brandon, my father's brother. They're not supposed to have statues, that's only for the lords and the kings, but my father loved them so much he had them done."
A Game of Thrones, Bran VII
Brandon the Daughterless
Another legend, amongst the Free Folk, of a Lord Brandon Stark.
"This was Brandon the Daughterless," Ygritte said sharply. "Would you hear the tale, or no?"
He scowled. "Go on."
"Lord Brandon had no other children. At his behest, the black crows flew forth from their castles in the hundreds, but nowhere could they find any sign o' Bael or this maid. For most a year they searched, till the lord lost heart and took to his bed, and it seemed as though the line o' Starks was at its end. But one night as he lay waiting to die, Lord Brandon heard a child's cry. He followed the sound and found his daughter back in her bedchamber, asleep with a babe at her breast."
A Clash of Kings, Jon VI
Brandon of the Bloody Blade
Another legend Brandon where Brandon the Builder is said to have descended from.
The list is long, and many are the legends, for there is scarce a noble house in all the Reach that does not boast of descent from one of Garth Greenhand's countless children. Even the heroes of other lands and kingdoms are sometimes numbered amongst the offspring of the Greenhand. Brandon the Builder was descended from Garth by way of Brandon of the Bloody Blade, these tales would have us believe, whilst Lann the Clever was a bastard born to Florys the Fox in some tales or Rowan Gold-Tree in others.
The World of Ice and Fire, The Reach: Garth Greenhand
Brandon the Bad
Their footsteps echoed through the cavernous crypts. The shadows behind them swallowed his father as the shadows ahead retreated to unveil other statues; no mere lords, these, but the old Kings in the North. On their brows they wore stone crowns. Torrhen Stark, the King Who Knelt. Edwyn the Spring King. Theon Stark, the Hungry Wolf. Brandon the Burner and Brandon the Shipwright. Jorah and Jonos, Brandon the Bad, Walton the Moon King, Edderion the Bridegroom, Eyron, Benjen the Sweet and Benjen the Bitter, King Edrick Snowbeard.
A Clash of Kings, Bran VII
King Brandon Stark
Another Brandon and this one forced the stoneborn back onto Skagos and destroyed their naval power. Note that the quote also includes ninth of his name meaning there are even more who aren't commented on.
Though rarely seen off their island, the stoneborn once were accustomed to crossing the Bay of Seals to trade or, more oft, raid—until King Brandon Stark, Ninth of His Name, broke their power once and for all, destroyed their ships, and forbade them the sea.
The World of Ice and Fire, The North: The Stoneborn of Skagos
I'm sure there are even more historical Brandon's but I think you get the idea by now.