In Kyon Big Damn Hero, members of the SOS Brigade change the way they address each other as a symbol of their True Companions relationship is strengthening: Tsuruya is called Tsu-chan by Haruhi and Mikuru(big), and Yuki and Mikuru being called by their first names not only by Haruhi.
Used in the The X-Files / Earth: Final Conflict crossover "Apologia." The young Agent Sandoval asks if he can address Scully as "Dana," because he refuses to give the impression that he's trying to take Mulder's place.
A classic example of this trope appears in With Strings Attached when, as a sign of his complete Heel-Face Turn, the Hunter asks Paul to start calling him by his first name, Jim. He had already agreed to use the first names of the four that morning, rather than the insulting nicknames he'd come up with for them at the beginning. (Originally, when George said that they have names and would he care to hear them, he said “Not especially.”)
In the Katawa Shoujo fic From Shizunes Perspective, Shizune temporarily switches to calling Emi by her first name rather than "Miss Ibarazaki" to make a point, then despite switching back to calling her by her last name, gradually switches over to her first name. She also points out that Hisao might be "interested" in Emi if he calls her by her first name, causing Misha to react in displeasure because she is in love with Hisao. Shizune also calls Misha by her first name, Shiina, in this fic, instead of her nickname.
Nav from Diaries of a Madman tends to be on first name terms with almost everyone he meets, royalty and commoners alike. Strangely enough, it's usually those accompanying him that protest over the lack of formality, rather than the recipients.
Discussed in "Aen'rhien Vailiuri". Tovan tr'Khev tells Jaleh Khoroushi to call him by his first name, because Morgan t'Thavrau, his neighbor and friend for ten years on Virinat and now his captain, doesn't do that anymore and he misses it.
In Out Of The Dead Land, Bucky initially switches back and forth between referring to himself and others in his head by their codenames or by their first names depending on whether he's trying to be the Winter Soldier or Bucky Barnes at that particular moment. As the fic progresses and he begins regaining his sense of identity, he eventually transitions to using first names only. Significantly, he does this first with Steve.
In Shatterheart, Kurogane usually refers to Real!Syaoran as "kid," but when he and Real! transition from friends to Friends with Benefits Kurogane makes a point of calling Syaoran by his name when they're alone. Kurogane temporarily rescinds this when he ends their relationship. He goes back to using it when he and Syaoran rekindle their bond.