Fëanor’s obsession with preserving his mother’s name reads pretty easily to me as stemming from his need to be in control of his mother’s legacy, and, consequentially, how he himself is perceived.
Míriel’s death and refusal to be reembodied, and Finwë’s subsequent remarriage, were highly publicized events in Valinor when they really ought to have been private - both because Finwë is king and his family life is always exposed to the public gaze, and because the situation with Míriel, Finwë, and Indis is completely unprecedented. No one has ever died or been remarried in Valinor, and the societal rules about this are so strict that the Valar feel the need to debate whether this is even permissible. As such, the situation must have been highly controversial among the Noldor - in the text, it’s presented as though Fëanor is the only one to object to his father marrying Indis, but in reality there must have been many supporters of Míriel who saw the marriage as an insult to their first queen.
On the flip side, there ought to have been a fair amount of people who placed the blame for the situation on Míriel. A society in which there was no death wouldn’t be very well equipped to deal with suicidality, and there must have been a fair amount of sentiment floating around claiming that Míriel was selfish, or weak, or an unfit mother and wife. But mentions of Míriel grew more and more taboo as the time passed. The concept of death in paradise makes people uncomfortable, and most people were content to smooth the situation over by pretending that Míriel had never existed - or that if she had, she was a brief stain on the king’s life, a rather embarrassing detail, like a mad wife in the attic.
So Fëanor grows up in a society in which the circumstances of his mother’s death - a source of significant trauma to him - are considered public property, and open to whisperings and speculations. On top of that, he grows up in a mixed household with children from two marriages - a complicated enough set-up in our world, and even more so in a society in which this kind of familial structure is completely unheard of. He understandably perceives his father and Indis raising a nuclear family as a denial of his mother’s existence of significance. And he knows that his mother died because his birth drained her of her energy and will to live, and feels a massive amount of guilt because of it. He fiercely dedicates himself to being the best he can be because he feels that the onus on is on him to make his mother’s death meaningful.
So his fixation on Míriel’s name might seem silly or fanatical or pedantic on the surface, and to a degree it is - it’s a very tiny detail to fixate on. But it’s representative of the larger social context of how his mother has been treated. By insisting that she be referred to by her preferred name, Fëanor is, on a micro level, trying to grant his mother the agency and self-determination that public opinion has denied her. And he wants to forcefully remind people of his mother’s existence and her importance to his own identity.