Present Mic isn't as oblivious as most assume him to be. Sure, he's loud and is over-dramatic about everything (yes, Shou, he is well aware of that he does that), but really all that is a front that he puts on. That's not really who he is. He sees the signs in his favorite student: the flinching, the overly cautious way he walks, how he can blend into a crowd even with his unique hair, the way he can erase his presence, the way he self-sacrifices over the littlest thing, how hard he works to try and please everyone, how he has hidden school supplies all over the school, has several copies of the textbooks, and most importantly the ever returning smile.
Now, the smile might not seem all that important, it might even seem like a good thing! ...Except Present Mic (aka Yamada Hizashi) is an expert on those smiles, the ever returning smile to hide your real emotions, afraid of what might happen if their real colors/emotions are shown.
Midoriya just screams abuse victim, which makes no sense. He has a powerful quirk that doesn't hurt others; he should have grown up praised and become more egotistical, not a depressed, self-deprecating teenager with loads of mental issues to work through. Yamada will admit that he is confused, he had hoped that Midoriya would realize how similar they were and try going to him for help, but even after the USJ incident, he has shown no signs of looking for help. (Maybe his mask is on too tight for Midoriya to see behind it? Or maybe he doesn't because he doesn't expect for him to find a kindered spirit?)
It was only after the Stain incident, that Yamada can't stop himself from reaching out to Midoriya, he has to try. Midoriya was going to kill himself at this rate and Yamada felt heartbroken over the mere thought. He doesn't wan to have to come to school one day to find another forever permanently empty when he could have done something to prevent that outcome from ever occuring. You see, Midoriya has become Yamada's favorite student, he is nearly fluent both in writing and speaking (just with an accent) English, so there wasn't much to teach him (though it was cool to find out that he was also fluent in sign). Although because of that little fact, Yamada was happy that it gave him an escuse so that he could constantly give him little candies whenever he got a perfect score (or just whenever he looked particularly sad) and get away with it, which was fairly often.
So when Yamada hears word of the Stain incident, he breaks down (he was really glad Aizawa was right there with him, he understood who he was under his loud personality, one of two -the other being his therapist) and decided to finally something for the little green bean, he can't wait for him to reach out forever. Seeing as Midoriya still has the majority of his internship left, Yamada gets Aizawa and Nezu's permission to let him intern under him at his agency (SpotLight Shows) for the rest of the break so long as he agrees to not have him partake in any actual patroling or be left alone for too long a period of time. (He does agree)
Midoriya readily agrees Present Mic's offer. It is only three days into Midoriya's new internship, when a problem arises. Yamada had promised his listeners, on his radio channel, that there would be a surprise guest, but right before the guest (pro hero Fat Gum) was set to arrive, one of his sidekicks called, saying that they were going to have to cancel because of a villain attack.
In a moment of desperation, Midoriya offered to be his guest which Yamada easily agreed to. That turned out to beone of the best decisions he has ever made. His radio Listeners loved Midoriya and fanboy nature. His hero analysis that he did on the spot becomes a monthly thing where Mic has Midoriya call in and analyze a pro hero, well known villain, or attack from earlier that day.
Of course after the internship, Yamada pulls Midoriya aside often to have him help with his classes or just to hang out or train. Eventually secrets were spilled (Ofa, bullying, overprotective/negligent mom) and they grow even closer bonding in a way that only true family members can. (If he ends up moving in with Yamada -and Aizawa- then it isn't mentioned at school.)
Plus Yamada conviences Midoriya to finally go to therapy; they end up sharing therapists.
All in all, by the time Midoriya becomes a pround member of the Class 2A, Todoroki has a theory of Midoriya being the secret love child of both All Might and Present Mic, with Midoriya Inko as the surrogate.