Don't worry! I am here! I've always wanted to make this essay about Izuku's name!
I don't think AFO necessarily gave Izuku his brother's name. However, I do believe that Izuku's name is actually a combination of both AFO's name and his younger brother's name.
You see, the kanji for Izuku's name is 出久. It's considered an unusual name, but still viable enough to be a name for a character.
The second kanji from his name actually came from his father (久 is pronounced "Hisashi" on its own)
There are actually characters whose names are a combination of their parants. "Katsuki" is an example of that. His name is 勝己, which is actually a combination of his parents' names. The first kanji came from his father (勝 is pronounced "Masaru" on its own), the second kanji came from the second kanji of "Mitsuki" (光己). It's also the same with Kyouka's name.
This is where Midoriya's name actually becomes a little suspicious for me...
Because Midoriya may have the kanji of his father for his name.... He somehow doesn't have the kanji from his mother...
The kanji for "Inko" is 引子, which is definitely not used in 出久.
Izuku is the only character so far, probably in the whole series, to have a name only coming from one parent. It'd make sense if his name has 1 kanji, but noooo. There's probably others like him, since we haven't met all the parents for the other characters. But Midoriya's the only one who has this confirmed so far.
Since I've been a huge fan of Dad For One theory, it doesn't seem farfetched that mabe AFO actually just decided to use his younger brother's name... and in my opinion, the name is actually fitting for the younger brother's name.
出 is pronounced "Shutsu" when read on its own. Though it can literally translate to "Coming out", and the kanji is often used for suffixes.
Some words that uses this kanji can sometimes roughly translate to "ORIGIN". "Shutsu" doesn't have a number that shows he's the First User, but it can indicate him being the original user of OFA. Also, the kanji is also used as a suffix for words that can mean "Rising of the sun/moon"
Also, this name stuff actually creates an interesting connection between Izuku and Tomura's names....
While Izuku ended up inheriting the first two names of the two brothers (Shutsu & Hisashi), Tomura ended up inheriting their last name (Shigaraki)
Honestly, I could be wrong. The name "Shutsu" might be too literal for a name, so it might be something else. But this is the headcanon I'm holding on to until canon proves me otherwise.
Also, Shutsu goes well with the younger brother's soft vibes. Just sayin'
