Did Lyanna Mormont ever ask Jon Snow to return the sword, Longclaw, to her family?

Question

Lord Commander of the Night's Watch, Jeor Mormont, gave his sword to Jon Snow. The sword, Longclaw, was a family heirloom. As a sword made of Valyrian steel, it must have been a very precious heirloom. I am surprised Jeor Mormont would give such an extraordinary item to Jon Snow even though he was grateful for Jon saving his life. He could have left the sword as an inheritance to his nieces or nephews.

There are only 200 such swords known in all of Westeros. Even the wealthy Lannisters tried to buy a Valyrian steel sword, but not even a minor house would not sell such a precious item to the Lannisters.

When Jon Snow met Lady Lyanna Mormont, she might have recognized the sword. She might even insist that Jon Snow return it.

Did Lyanna Mormont ever ask Jon Snow to return Longclaw to her family?

If you have an answer from the books, I will accept that over answers from the TV shows or mere speculation.

Answer 1

Jon Snow does not meet Lady Lyanna Mormont in the books. If such a meeting is going to occur, it will happen in the sixth book which isn't published yet. There is no explicit mention of Lady Mormont asking for it in the show. Also, given that Lady Mormont isn't physically large enough to wield it herself, I think it would be reasonable to guess that she would allow Jon Snow to use it until she was bigger even if she did want it back. Valyrian steel, obsidian, and fire are the only known effective weapons against the white walkers and as you said, valyrian steel is in very short supply.

In addition, in the books Lyanna Mormont is only regent of Bear Island, her mother Lady Maege Mormont is as far as we know, searching for Greywater Watch in the marshes of the neck, with Lyanna's older sisters Lyra and Jorelle. Lady Maege is Jeor Mormont's sister. The heir of Bear Island is Alysane Mormont, who is escorting "Arya Stark" (actually Jeyne Poole Sansa's childhood best friend) to Castle Black on Stannis Baratheon's orders. If any of the Ladies Mormont are going to ask for Longclaw back, Alysane will be the one with first opportunity to do so.

Answer 2

House Mormont with Lyanna at his head are very firm in their support of House Stark and by extension Jon Snow.

Lyanna Mormont: House Mormont has kept faith with House Stark for 1,000 years. We will not break faith today.
_{Game of Thrones, Seaosn 6 Episode 7, "The Broken Man"}

Also Lyanna is still young but she is incredibly smart for her age. As such I believe she recognises that the sword will do more good to Jon than it would to her. She's possibly too small to actually wield the sword effectively yet as well.

When the Old Bear gives Longclaw to Jon he also recounts that he's had the pommel changed from a bear to a wolf. As Jon hasn't unsheathed the sword in front of Lyanna it's possible she doesn't recognise it too.

Jeor Mormont: Good. You'll be ready for this, then. l thought a wolf was more appropriate for you than a bear so l had a new pommel made. lt's called Longclaw. Works as well for a wolf as a bear, l think.
_{Game of Thrones, Season 1 Episode 9, "Baelor"}

The sword was also given to Jon from the Old Bear and as such House Mormont probably recognises it as a official change of hands, if they know about it. The sword was also meant for Jorah but when he brought dishonour on the house it was left without an owner.

Jeor Mormont: lt was my father's sword, his father before him. The Mormonts have carried it for five centuries. lt was meant for my son Jorah. He brought dishonor to our House, but he had the grace to leave the sword before he fled from Westeros.
_{Game of Thrones, Season 1 Episode 9, "Baelor"}

Lastly, note that Lyanna is actually the niece of Jeor and is not of the main branch of the house, though through exile/death/Night's Watch her branch has become the leaders of House Mormont. As such the sword never would have normally passed to her but to Jorah.

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On a side note in Season 7 Episode 6, "Beyond the Wall", Jon offers the sword to Jorah but he declines it stating that it is now Jon's sword after he forfeited his own claim. This also points out that maybe Jon didn't offer it Lyanna because he still believed Jorah to be it's rightful owner.

Jon Snow: Your father gave me this sword. He changed the pommel from a bear to a wolf, but it's still Long Claw. Lord Commander Mormont thought you'd never come back to Westeros. But you are back and it's been in your family for centuries. It's not right for me to have it.
Jorah Mormont: He gave it to you.
Jon Snow: I'm not his son.
Jorah Mormont: I brought shame to my house. I broke my father's heart. I forfeited the right to claim this sword. It's yours. May it serve you well and your children after you.
_{Game of Thrones, Season 7 Episode 6, "Beyond the Wall"}