Brendan

Brendan (T.) Birch is the son of Professor Birch, as well as A's next door neighbor and rival. Some fan art so far has suggested A has a relationship with him that, depending on the canvas section sampled, swings from violent infatuation to a more benign if unconventional friendship (sometimes bordering on flirtatious behavior) with him that may or may not be reciprocated.

He has been christened Mr. T by some for his nickname, named after one of the Lost Characters. In early game lore, it was theorized by people who believe Camila A. Slash suffered from DID, that T/Brendan was the only escapee from A's (or perhaps Slash's) pre-game crimes, and is either pursuing her for revenge or justice, or watching over her and spying on her for some reason. This theory was only mentioned in the first few days of the run, never got popular and was never followed up upon, especially after the murderer lore surrounding A was largely dropped.

A more conventional view, shows Brendan as simply being another trainer who periodically meets up with A as her somewhat dorky next-door neighbor and friend. He's a bit of a doormat with a rather unfortunate childhood, and helps A out by giving her certain items she needs to advance, even though she gives him a hard time and makes fun of him, often physically hurting him in the process.

He travels around with a Marshtomp called Oscar. His goal is merely to do some field work for his father. His disinterest in earning badges is explained in fanart by this not being his first journey around Hoenn and him having earned them all already at an earlier age. The Mudkip that's seen with him during the TPP run is the offspring of his actual starter from back then, Swampert.

The Life and Wiles of Brendan T. Birch
Most of the lore surrounding Brendan stems from a comic series hosted on the TPP subreddit called "The Life and Wiles of Brendan T. Birch", drawn by reddit user /u/Everyle. It revolves around the things Brendan was up to during the TPP run and ends in a comic arc explaining his childhood and his relationship with A. Starting out as funny gag strips, the comic slowly grew more serious and changed in style to match the subject matter. A key element to the story is the childhood trauma Brendan experienced by hand of his mother, who wanted a girl rather than a boy and didn't accept him as her child. This reflects the extreme desire of the TPP players for a female player character during the crystal/emerald intermission, and people suggesting calling the character "Brenda" and forcefully making him trans if a male trainer was chosen, which ultimately ended up not happening. A link to the comic archive can be found here

During the Heartgold run, a parody of The Life and Wiles of Brendan T. Birch popped up on the TPP subreddit, created by a clone account called /u/Everyie. These sprite comics parodied pages of the original series with purposely misspelled dialogue and crude ms paint edits, depicting Brendan as the personification of the word swag, often seen with shades and a Mountain Dew cap. A link to the compilation post can be found here

Battles against A
A's first battle against Brendan on Route 103 ended up in failure after she constantly had Zexy use growl on his Mudkip instead of an attacking move. She defeated him on her second attempt at 0d 2h 11m.

Trivia

 * Brendan is the first rival to have defeated the player character on their first battle.