Felicity's first thought when meeting John Diggle (probably most people's first reaction, too) is that the guy is huge. With his height and muscle, he manages to make Oliver look small- which is no mean feat, considering that Oliver is pretty built himself. Her second thought is holy shit he's ex-military. Diggle probably knows multiple ways to kill her with paperclips, or whatever else they teach combat specialists these days. Overall, the effect is pretty intimidating. And Felicity's not afraid to admit that she's a little scared, working with two really dangerous people in a secret lair underneath a nightclub in the most dangerous part of town.
That changes, though, when Felicity has her third big thought about John Diggle: that he's a godsend. Because, behind that giant physique is a heart just as big to match, and it's always ready to support Felicity when she needs it. Diggle is a salt-of-the-earth kind of guy; always calm, always practical, loyal to a fault. He sees the truth of the world around him and reflects that verity onto other people. Whenever Felicity is panicking because she's new to the whole high stress life or death thing, and Oliver is being stubborn and, well, Oliver, Diggle is the one who tethers them to earth. Somehow, he knows that Felicity responds better to questions and goals rather than pointless platitudes; he can tell when Oliver is having a hard time and needs someone to talk to. His steady competence is invaluable.
The fourth thought that Felicity Smoak has about John Diggle is that he's far more complex than many give him credit for. Diggle's solid, steady, but not staid. As she gets to know him better, Felicity finds quirks and qualities that stay hidden from the casual observer behind a mask of stoicism. She discovers that Dig's got a good sense of humor. He's always ready with a sarcastic quip that he'll deliver, straight-faced, to Oliver when he's being ridiculous, or a small self-depreciating smile when he does something silly himself. She learns that when he gets curious, Dig's as insatiable as she is- he'll get giddy and excited and very un-Dig like and ask a million questions. She discovers that they share a deep appreciation for fried, unhealthy foodstuffs and science fiction. Sometimes, when Oliver is off being broody and mysterious, she and Dig will sit in the Arrow-cave with milkshakes and Big Belly Burgers and argue about which Star Wars film is the best, or whether the Terminator reboot is going to be any good. When Dig manages to find a babysitter for Sara, he texts Felicity about which tie he should wear to his dinner-date with Lyla; when Felicity considers joining a gym Dig gives her recommendations. They're co-workers, friends, and allies against the demons of Oliver's past. They share a bond based on trust.
But then, Felicity shatters it. She lets the bad guys get away because she is terrified of losing more friends, and she can see the betrayed hurt on Diggle's face. So she leaves, she withdraws, because they can be stupid and get themselves killed but they can't make her watch; can't make her analyse every detail of their dead bodies like she did for Sara. Oliver's gone, Oliver's dead, and they're all doomed without him.
Later, realization sparks. She had always thought that Oliver was invincible; that he would always manage to cheat death like a modern-day Robin Hood. Despite all logic, she had believed in the mythos surrounding his superhero persona. But, that had never been true. Oliver was skilled, and incredible, and amazing at what he did- but he was still just a person, in the end. Just like she, and Roy, and Diggle were people. Oliver had dedicated himself to a cause, but the cause didn't have to die with him. And, ultimately Oliver didn't die for the city; he died for his sister. It makes sense, to Felicity. By taking up the mantle, they can honor how Oliver lived and how he died. They can pay homage to his dedication to justice, to Starling, and to protecting the people he loved. They can go on being Team Arrow.
When she pitches her idea to the boys, Felicity is passion and vision and everything that she'd been missing since Oliver died. But, the whole time her gaze drifts to Diggle in particular. 'Please, forgive me', she implores in her mind. And then, Diggle agrees, and she can almost feel the nebulous idea of them solidifying into a group, because he's always been a cornerstone. It's creation, and moving on from grief, and forgiveness.
The next weeks are filled with bumps as they adjust, and although much of it is worrying and tense it's also so rewarding because Felicity is rebuilding her family of superheroes and misfits. She can see that Diggle is beginning to trust her again, and they start to regain their old comfortability.
And when Oliver comes back, things are different. Felicity has changed so much and has re-evaluated her (unrealistic) views of Oliver, and talking to him seems to be like talking to a stranger. It would all be very upsetting, but Felicity knows that she can survive it, because she has before... and because, once again, she has Diggle's strong support behind her.
It may have started out as Oliver's personal crusade, but now it's their collective mission; and Felicity and Diggle will fight to keep it on a moral course. She knows that they can do it, because they've been there from the start, she and Diggle, and together they can move mountains.