In other words, best princess. XP

(This image isn't mine.)

What makes her likeable? Plenty.

1. Her fantastic design and colors. Of all the ponies, she is the one that stands out the most. With most of the others being bright and pastel, her colors are very pure and mostly dark. Almost every single color is a shade of blue — mane, fur coat, shoes, and cutie mark. By blending the colors together, she looks very calm, cool, and quiet. Combined with her jade eyes (which represents wisdom and generosity), the incredible balance in her character design makes her both regal and approachable.
   
2. She has the status of underdog. In the Western world, underdogs are admired. People want to see the underappreciated succeed. Princess Luna's past is very relatable, because many of us have been in situations just like hers. She wants to live beside Celestia, but she was honored much more than Luna, resulting in the jealous rage that caused her to spend a millennium in prison.
   
3. She's adorkable. Twilight isn't the only character to look adorkable (which, BTW, is now accepted in the Collins dictionary).
    
   
   
4. She is a three-dimensional character with a very mysterious beginning. Part of her popularity stemmed from after she was reverted back to her normal state, but she didn't appear again for the rest of the season. Therefore, the fandom started to fill in ideas that helped plug in many facets of their own identities onto her, some of which came true in Luna Eclipsed, her breakout episode. In LE, she tried the best she could to adapt to current society, but was unable to because her past mannerisms are archaic and scared so many ponies. Throughout Nightmare Night, she had to grow and become accustomed to Ponyville and all of Equestria today. With the help of Twilight Sparkle, she loosened and had fun, including giving Shadow Dash some much-needed payback.
   
5. At the same time, she still tries to prove her worth. One of the littlest, yet most important details of her character came during the first act of The Crystal Empire, Part 1. As the two sisters stared out the window, she wanted to volunteer in helping keep the revived Empire safe. Just through that conversation, she obviously felt she had unfinished business and wanted to reestablish her worth as Equestrian princess the right way.
    
   This was backed up during the beginning of Princess Twilight Sparkle, Part 1. As they watched Luna raise the moon, Celestia and Twilight conversed the following:
    

   
   Princess Celestia: For my subjects, it has always been a celebration of my defeat of Nightmare Moon. But for me... [sighs] It was just a terrible reminder that I'd had to banish my own sister.
    
   Twilight Sparkle: I guess I never really thought about it that way.
    
   Princess Celestia: But now it has become a wonderful reminder of her transformation back into Princess Luna, and our happy reunion. I am so pleased that you will be playing a role in the festivities. I know it must have been difficult to see your friends return to Ponyville without you.
    
   Twilight Sparkle: Maybe a little.

6. She actively leads to prevent others from walking down her path long ago. In Sleepless in Ponyville, Luna told Scootaloo to talk to Dash about her insecurities and ask if Dash can mentor Scootaloo. But that wasn't the only time she went into nightmares. She connects her journey as Nightmare Moon to Sweetie Belle in Toils and tells her to not let her jealousy-induced rage influence her to doing something she may forever regret.