Westeros has produced some great and some questionable monarchs. One dynasty that left a lasting and burning impression was that of the Targaryens. Despite the numerous Targaryens who ruled the Seven Kingdoms, there were many others in the family who didn't get a chance to prove their worth as rulers even though by all accounts they would have made better kings and queens than some of those who achieved the throne through birthright.
As Cersei Lannister famously said, "Every time a Targaryen is born, the gods flip a coin." She referred to the trait of insanity that frequently plagued Targaryens, something viewers will continue to see more evidence of as House Of The Dragon unfolds in future seasons.
Baelon Targaryen
Father of Viserys I and Daemon Targaryen, Baelon was known to be a favorite among the smallfolk and commonly referred to as Baelon the Brave. Although he was the heir to the throne after the death of his elder brother Aemon, he was outlived by his father, causing confusion and debate as to who should next sit on the Iron Throne.
Baelon was strong, often showing courage along with being substantially more battle-headed than his older brother. His strength in avenging his brother's death by riding on Vhagar to Tarth in order to fight the Myrish exiles that had slain his brother saw him hailed as a hero in King's Landing. Being popular with the smallfolk would have served Baelon well had he of been king, he would have had the support of the nobility as a headstrong male with a clear claim.
Viserra Targaryen
As the tenth child of King Jaehaerys I and Alysanne, Viserra had a very slim chance of inheriting the Iron Throne. Along with the fact that she was a woman, she was the sixth daughter born to Jaehaerys and Alysanne Targaryen so was unlikely to even be made a sister wife to heirs Aemon and Baelon.
For someone so far down the line of succession, Viserra's ambition and will is something to be admired, she was fully aware of her own beauty and the power she had over the simple men that fawned over her. Given the chance, Viserra could have used her ambition and attractiveness to steer reform within the Small Council. She stopped at no efforts to seduce her brother Baelon after the death of his wife as a ploy to be his consort, her decisiveness and clear mind on what she wanted could have been an asset to the Seven Kingdoms.
Saera Targaryen
Arguably the most formidable of Jaehaerys and Alysanne's children, Saera was the ninth child born to them and beheld most, if not all the great qualities that their other children possessed. However, she was also incessantly stubborn, definitely holding the idea that Targaryens were closer to gods than men in high regard.
Saera had a clever mind and sharp wit, along with this her classic Targaryen beauty made her popular. She was always ambitious to become a queen, and was able to easily manipulate most people into finding her charming and getting them to do her bidding. When Jaehaerys called a Great Council, Saera possessed the greatest claim to the throne, although she was not married and her children were bastards, it would have been undeniable for the lords of Westeros that she held the strongest claim to the Iron Throne. Given the throne, Saera would have been decidedly more liberal than rulers before her, especially considering her affiliation with pleasure houses.
Aemond Targaryen
As the second son born to Viserys I and Alicent Hightower, Aemond was not intended for the throne. He held the grudges passed down from his mother, and was underhanded in attacks he made in the name of his house.
Aemond was a talented swordsman, and he even attempted to enact diplomacy by negotiating and persuading House Baratheon to support the claim of his brother Aegon II during 'The Dance of Dragons'. His strong-willed nature would have made him an effective ruler, changes would have been swiftly made, and the crown would have arguably been restored to some of the glory it had lost as Aemond often took what he believed belonged to him and his family, whether that be a territory or a debt.
Rhaenys Targaryen
As the firstborn and only child of Aemon Targaryen and Jocelyn Baratheon, Rhaenys was set to inherit the Iron Throne. However, the premature death of her father led to her cousin Viserys being favored as heir, as the eldest son of her uncle Baelon.
Despite being passed over in favor of her cousin, Rhaenys did not take up arms against him Instead, she accepted his rule and did so graciously. As a realist, Rhaenys knew that the lords and smallfolk better preferred a man on the Iron Throne, however it is this realism that would have served her well as a ruler. She was a clever individual, and often referred to as 'The Queen Who Never Was'. Rhaenys had the skills and the upbringing to rule Westeros, but not the correct gender.
Jon Snow (Aegon Targaryen)
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The only child of Rhaegar Targaryen and Lyanna Stark, but raised as the bastard son of Ned Stark. Jon had just as much, if not a stronger claim to the Iron Throne than Daenerys Targaryen, who was his aunt as well as lover.
Known to be loyal and just, Jon Snow had all the qualities you would want in the Ruler of the Seven Kingdoms. As well as experience in diplomacy, he was a trained swordsman who fought not just for his own cause, but for the right ones. A champion of those less fortunate than him, Jon would have made an amicable, yet strong leader. As a male with a strong claim to the throne, he would have been readily accepted by Westerosi lords.
Daena Targaryen
Eldest daughter of Aegon III and commonly referred to a Daena the Defiant, Daena Targaryen was considered as heir to the Iron Throne, but was cast aside due to her gender and lack of allies.
Known to be smart, a great huntress and beautiful, Daena was the most popular of the three daughters of Aegon III. Her status is confirmed by the fact she is considered for the role as queen, which is primarily dismissed due to her lack of allies after spending around a decade locked away in the Red Keep with her sisters. Had she of been given the chance, Daena would have proved a willful and clever ruler, along with most likely having the respect of most of the lords that occupied her court.
Aemon Targaryen
The eldest surviving son of Jaehaerys I and Queen Alysanne, Aemon was first in line to the Iron Throne and expected to make an excellent ruler as he was already popular as the Prince of Dragonstone.
Aemon was known to be quieter, and had he of ascended to the throne, would have been one to listen to the council of the lords. Unlike his younger brother Baelon, Aemon was not as quick to fight or judgment, which is likely a better quality for a king. Had Aemon sat on the Iron Throne, there would have been no question whether Rhaenys would have inherited it. He was also brave in battle, which would have boded well with all of his subjects, but ultimately it led to his death at the hand of Myrish Pirates.
Daeron Targaryen
Commonly referred to as Daeron the Drunken, he was the eldest son of Maekar I and in line to be king. Not wildly popular but not disliked, Daeron was outlived by his father, causing another Targaryen succession crisis.
Daeron had prophetic dreams, like some of his ancestors, which may have been considered yet another symbol of Targaryen madness. Despite his drunken reputation, Daeron's prophetic dreams could have proven instrumental had he of been given the chance to rule Westeros. As he did not possess an affinity for fighting, Daeron would have likely been a peaceful king who spent his time building up the realm's defenses and coffers, which may have been just what Westeros needed.
Maester Aemon
While alive, Maester Aemon was believed to be the last male Targaryen, the son of Maekar I and Dyanna Dayne. He was unable to ascend to the Iron Throne as he had sworn to be a maester.
Maestor Aemon was exceptionally wise and learned and demonstrated clear loyalty. He had a strong sense of fairness that was frequently displayed when he was a maester of the Night's Watch. Although he could not have sat on the Iron Throne because of his vows, Aemon would have made a clear-headed ruler, and his ascension to the throne likely would have prevented the Targaryen dynasty from being overthrown.