Historical Half Targaryens with Non-Valyrian features
That's the case with many Half Targaryens in history.
King Daeron II married Mariah Martell. His first born son, Prince Baelor had dark hair of his mother.
Of Prince Maekar he saw no sign, but he recognized Prince Baelor at
Lord Ashford's side. Sunlight flashed golden off the shoulder clasp
that held his cloak and the slim coronet about his temples, but
otherwise he dressed far more simply than most of the other lords.
He does not look a Targaryen in truth, with that dark hair. Dunk said as much to Egg.
"It's said he favors his mother," the boy reminded him. "She was
a Dornish princess."
Dunk and Egg: The Hedge Knight
Prince Baelor's own sons Valarr and Matarys had the same hair.
King Daeron's fourth son, Maekar had Targaryen silver hair. But he married a Dayne. So his first born son, Prince Daeron also had the common hair instead of the silver-gold hair of Targaryens.
Dunk chewed, swallowed, and tore off some more bread. "Why did you do
it? Was it some jape, to make a fool of the stupid hedge knight?"
"No." The boy's eyes filled with tears, but he stood there manfully.
"I was supposed to squire for Daeron. He's my oldest brother. I
learned everything I had to learn to be a good squire, but Daeron
isn't a very good knight. He didn't want to ride in the tourney, so
after we left Summerhall he stole away from our escort, only instead
of doubling back he went straight on toward Ashford, thinking they'd
never look for us that way. It was him shaved my head. He knew my
father would send men hunting us. Daeron has common hair, sort of a
pale brown, nothing special, but mine is like Aerion's and my
Dunk and Egg: The Hedge Knight
Similarly, Ser Harwin Strong fathered three bastard sons on Princess Rhaenyra Targaryen , all of them inherited their real father's brown hair (But supposedly their legal father was Laenor Velaryon, who didn't care about getting cuckolded).
Whatever the truth of these tales, it was soon announced that the
princess was with child. Born in the waning days of 114 AC, the boy
was a large, strapping lad, with brown hair, brown eyes, and a pug
nose (Ser Laenor had the aquiline nose, silver-white hair, and
purple eyes that bespoke his Valyrian blood). Laenor’s wish to name
the child Joffrey was overruled by his father, Lord Corlys. Instead
the child was given a traditional Velaryon name: Jacaerys (friends and
brothers would call him Jace).
The Rogue Prince
Princess Rhaenyra had given birth to a second son late in the year 115 AC. The child was named Lucerys (Luke for short). Septon Eustace
tells us that both Ser Laenor and Ser Harwin were at Rhaenyra’s
bedside for his birth. Like his brother Jace, Luke had brown eyes
and a healthy head of brown hair, rather than the silver-gilt hair
of Targaryen princelings
The Rogue Prince
In 117 AC, on Dragonstone, Princess Rhaenyra bore yet another son. Ser
Laenor was at last permitted to name a child after his fallen friend,
Ser Joffrey Lonmouth. Joffrey Velaryon was as big and red-faced and
healthy as his brothers, but like them he had brown hair, brown eyes,
and features that some at court called common. The whispering began
again. Amongst the greens, it was an article of faith that the father
of Rhaenyra’s sons was not her husband Laenor, but her champion,
The Rogue Prince
Other than that, King Aegon IV sired a ripe crop of bastards. Three of them were sired on a black courtesan, named Bellenora, Narha and Balerion. Their description is not given but since Bellenora came to be known as the Black Pearl of Bravos, I suppose she was black as were her siblings. He also had Aegor Rivers, better known as Bittersteel, the founder of Golden Company, with Lady Barba Bracken. The lad had purple eyes from his father but inherited black hair of his mother.
King Aegon V married Betha Blackwood. His eldest son was a ravenhead. His physical description is not given but his illustration in TWOIAF shows that he inherited raven locks of his mother, unlike his brothers and sisters.
And most Recently, Prince Rhaegar's children with Elia Martell are good example. Rhaenys took after her mother and inherited her dark hair but Aegon took after his father. So among all three of Rhaegar's children, only Aegon took after him while Rhaenys and Jon took after their mothers.
Historical Half Targaryens with Valyrian features
While it can happen (Half-Targaryens taking after non-Targaryen parent), as we see evidence above, it is not something which always happens.
King Aegon V married Betha Blackwood, all of their children were your typical Targaryens except the eldest Prince Duncan the Small, Prince of Dragonflies (See above).
King Daeron II's all children were Typical Targaryens except Baelor Breakspear.
King Maekar's all children were typical Targaryen except Daeron Targaryen.
Princess Rhaenyra's mother was an Arryn (although Lady Aemma Arryn herself was a half Targaryen and cousin to her husband), but she was typical Targaryen. Rhaenra's half siblings Aegon II, Aemond, Daeron and Helaena were all typical Targaryens as well despite having a Hightower (Although Alicent Hightower's hair color is not mentioned) for their mother.
Case of Jon and Arya
So it shouldn't be surprising that when Rhaegar (Silver Gold hair) married Lyanna (Brown hair), their child was born with Brown hair as Silver-Gold is a recessive gene, while Brown is dominant.
Jon had the brown hair of Starks, which he inherited from his mother. Arya inherited the same from her father.
It would have been easier if Arya had been a bastard, like their half
brother Jon. She even looked like Jon, with the long face and brown hair of
the Starks, and nothing of their lady mother in her face or her
AGOT: Sansa I
They both also had the Grey eyes of Starks. While everyone wondered who was Jon's mother who left nothing of her in him, It was actually his father who left nothing of him in Jon.
In any case, Jon had a good enough chance of getting Targaryen outlook but his mother's genes won and he was born lucky, looking like a Stark, making it feasible for Eddard to lie. Otherwise explaining a Valyrian looking bastard would have been really hard. Would have gone something like this:
Robert: Say Eddard, how did you sire a Valyrian looking bastard?
Ned: I went to Lys.
Robert: When? You never had the time to go to Lys.
Ned: Okay you got me. I had an affair with Queen Rhaella.
Robert: Rhaella? Hmm I don't see the timeline match but she was with
child when she fled. But her Child was Daenerys Stormborn.
Ned: She's my daughter too. Jon's twin.
Robert: GODDAMMIT NED! TELL THE TRUTH!
- Why didn't Eddard tell Catelyn about Jon's true parentage?
- Is there any tangible proof of Jon's parentage?
- Timeline of Lyanna's elopement with Rhaegar
- Will Jon still be a bastard?