You can give details for some, like the reason why you changed the battletag, or why you picked it, or what the name means. As with nearly all my discussion topics, I’ll go first!
This is in order of first battletag ever to my current one
1. Mei-
I fell in love with Mei when I first bought the game, she was my first Mein. Something about her sweet caring demeanor just got to me. This was absolute peak season for hating on her though.
Every single game I was getting called satan and pure evil and heartless. It was really getting to me. People would blame me if we lost the game. I would get literally yelled at in mic if I made 1 mistake with my wall and blocked a team mate off on accident. I started to dread playing the game, but I loved the game so I was so conflicted.
I eventually wrote a support ticket to Blizzard explaining it all and, bless them, they gave me a free battletag change 
2. Starfire-
I chose this because I just really admire Starfire from Teen Titans. Plus 8 letter double-word names are really balanced and nice to me.
3. Delphi-
This is when I bought a 2nd account to do a pure literal Mercy only pacifist account (I unbound swap weapon and melee). I was going to challenge someone else for the world record in Mercy-only playtime.
I lost interest when masteroverwatch’s servers shut down, plus when the top 5 heroes chart was removed, the thing that showed everyone your highest playtimes on your mains for each match. I still play on here during the occasional blue moon but not as much as before
4. Athena-
For a very long time I have admired and revered the Greek Gods as a Pagan. Athena is my Matron (A matron is basically your primary goddess that you worship over all others in your life) (you might have seen patron/matron before or patron/patroness, this is the same thing, it means your primary god/goddess if you are in a polytheistic religion.
Of course nobody picked up on that and the people that commented on my name during this time simply talked to me like I was Winston’s AI haha
5. MeisThighs-
Not even gonna lie, I named myself this for a couple of months. I just wanted everyone to know I love Mei’s thighs. lol.
6. Taiga-
I swapped my name to Taiga after I watched Toradora last year and it became my number 1 anime and Taiga (the main girl love interest of the show) became best girl to me. Even if people don’t know about the show, it still sounds cool. At least 1 person has thought I meant Tyga the rapper though
She’s a big fan of that anime too. Think she’s on the forums as well, but goes by a different name here.