I apologize for the length (or lack of) of the headcanon.
Headcanon 20: Lin's snarky nicknames for Tenzin included: Baldy, Feathers, Twinkletoes Jr., and The Serious One.
From the time she learned how to talk, Lin had always been using nicknames against Tenzin, just as her mother had with Aang. Toph had taught her plenty of her original nicknames for the Avatar. In fact, Toph had taught her to call him Twinkletoes before Aang. It drove him crazy because every time he held her, Lin would screech out "Uncle Twinkletoes!" Lin then decided, if Aang was Twinkletoes, then wasn't Tenzin Twinkletoes Jr.?
Toph also called Aang "Airhead" and "Baldy" all the time, so Tenzin became Airhead and Baldy.
Though around Lin he was a bit lighter, everyone, especially Lin, called Tenzin "The Serious One." unlike his siblings, Tenzin was always so serious and levelheaded. However, when Lin was around, he was much less solemn, though it wasn't gone forever. Sometimes, when Lin would greet him and see the joy on his face from seeing her, she'd asked, "Why so serious, The Serious One?" Then he'd laugh and the smile wouldn't leave his face for hours.
"Feathers" was a personal nickname created by Lin. When they were kids, Lin and Tenzin had been engaged in a mass pillow fight. Lin chucked a pillow at him hard and it exploded all over him. Feathers stuck all over Tenzin and his robes, making him look like some sort of exotic bird. Lin had nearly died laughing as Tenzin spun himself in the air to try to free himself from the feathers. However, he was to no avail. "What's wrong, Feathers? Can't fly away?" She then helped him pull each and every feather off of him all while making fun of him.
Even though Lin taunted Tenzin with nicknames, never did he once call her anything other then Lin.