It's important to first look at is Jon's status when Jeor bequeaths the sword to him. Jon is bound to "bear no child." This is quite profound.
When Jon becomes old, he will likewise have to bestow the sword. He'll have to make a choice not based on biological lineage.
Will he bequeath it to the most worthy, as Jeor deemed him? Will that person be his successor in the Watch, and in that case, was it partly a gift to the Night's Watch?
What if Jon dies in battle but body and sword are recovered? Who does the sword go to? Is it returned to the Mormonts?
Everything changes when Jon is freed from his oath. He may now have children.
But would he bequeath it to his children, knowing Jeor never expected he would have them? After all, Jeor's gift of the sword to Jon broke the tradition of primogeniture. Jon refused the blade when Jeor first offered because the act carried such tremendous implications. [See "Blackfyre" below]
Dawn, the sword owned by House Dayne, is not necessarily wielded in every generation. Only those deemed worthy may bear it, and they take the title "Sword of Morning". This implies custodianship of a sword may skip a generation.
**Because Jeor bequeathed it on Jon, it would certainly be a breach of etiquette for Lyanna to ask for it back while Jon bears it. It may not be unreasonable to ask that it to be returned after Jon's custodianship is done. **
Jon, after all, has great admiration for Lyanna.
Blackfyre, the Targaryen sword, was not bestowed by Aegon the Unworthy on firstborn son. Rather, he gave it to his bastard, Damon. The act was so unheard of, it was taken as a sign of the Unworthy's intention that Damon should rule. This was the cause of the Blackfyre Rebellion.