(Edited for title — Tim)

Perhaps the Hulk's origin would not involve gamma radiation, but something else (super-steroids, genetic engineering, etc). Some characters' civilian occupations would be web designers, app designers, etc.

Peter Parker would receive an iPhone instead of a microscope!

Just throwing things out there, I am sure many will add to the list.

Although, having said that, I like how Marvel sidestepped that in the latest version of Spider-Man - everyone has seen one version of an origin for him so just get on with it.

I’m reminded how prominent mutants were in. The ‘80’s & ‘90’s where is seemed that every new character was pretty much a mutant, so origins seemed to become ‘I hit puberty’ & that was it. I’d still like to see more of Stan’s Origins - they were inventive, but told so quickly

We'd have a great many "garage heroes," characters whose powers are a result of their having labored on their own in obscurity, until Bam! Something amazing happens, not out of coincidence or sheer luck, but because they intended for it to.

Radiation wouldn't be a prevalent factor in super-heroic origins, but self-involvement and self-interest would. Satellites would often play a role, as would a deep-space robot probe or two. Mutants wouldn't gather together at one location, but would be linked through social media and become a force for social change, with widely polarized opinions on both sides of the aisle. The clash between Professor X and Magneto's opposing philosophies would be a constant flame war online. 

Aliens wouldn't travel about in rockets or saucers, but by means of teleportation and spiritual advancement. Crystals would be involved in many cases, and Skrulls wouldn't be here because they mean to attack us. They'd be here because we brought them upon ourselves with our negativity and obstructive spiritual practices. If we could get our karma straight, we wouldn't be magnets for these "bad aliens" to come and inhabit the Earth. We could be free and participate in the universal community instead of fearing it and fleeing from the representatives it sends and we draw from it. Everything is explained in my new 17-course online education program, "Ohm Invasion." 

And Captain America could very well have been a rescue worker when the towers fell who has been preserved in a secret government facility since the collapse. Why? What does he know? What secrets does he possess? Or was he just a good man a team of scientists felt deserved a chance of life for his good works? 

Conspiracy theories could run rampant throughout a modern, unprecedented Marvel, with countless villains heavily invested in "revealing" the "truth" about so many topics to a disbelieving public. Countless bad guys would be all about tearing the lid off the Moon landing, black helicopters, and jet contrails. Thor could be a representative of the Realm of the Flat Earth, which he insists is our Realm, a "fact" we must all recognize as the Ice Walls have begun to close in... 

Definitely stuff with robotics and cybernetics. I like the idea of people augmenting themselves in their "garages" or at a school 3d printer lab. Could also see freak accidents happening with augmentation that has unpredictable and cool results.

Also, likely still chances for dimensional weirdness and explorations of the far reaches of space and even the molecular level Tardigrade-girl anyone?

If Stan was doing this I think you'd see interesting approaches to:

Social media (like was said above and even explored in X-statix actually, where heroes were like athletes)

Bullying- different than the  bullying that Peter Parker took I imagine.

WAY more racial diversity of course which would be great.

Gun violence and other hot issues of the day would get looked at at least somewhat. An origin built around easy access to guns and the impact of that...

Fun topic!

I think Captain America would have the same updated origin.  Maybe it wasn't the Avengers who found him, but the "floating as an ice cube" scenario from WWII still fits today.

The Fantastic Four could still get their powers from Cosmic Rays, but how they got into space would likely be different.  Sue Storm wouldn't be just the girlfriend who tagged along in the rocket ship (along with her kid brother), but have some scientific rationale for being there (similarly for Johnny).

The Hulk could still have a similar origin, but maybe wouldn't be subjected to the radiation in the USA.  Maybe he got created through North Korea's testing instead.

Spider-man's origins can be the same.  His personality might be a little different, but being a science nerd isn't out of place now as it wasn't out of place in the 1960s.

Probably the most substantial update would be more diversity in who was getting transformed.  Stan and Jack primarily used WASP as their blueprints back in the 1960s, which reflected what was going on in their world  Now, there would be more racial differences, cultural differences etc. as to who was getting what.

Marvel already did this. It was called the Ultimates.

When Fantastic Four and Spider-Man started, they were so different and innovative that they basically started a new genre of comic books... human drama and situations. If that weren't around, could DC have made that transition? Would those editors and writers even recognize that market? And then, could they have written for them? (Bob Haney, try that he did, was no Stan Lee...)

The nature of the powers was never the impetus of Marvel; it was always the human stories with people with special abilities. Iron Man and the Hulk aren't quite the interesting aspect; Tony Stark and Bruce Banner are. But if, in some capacity, a wealthy industrialist were on an inspection tour of, say, a Chinese facility who was subject to a disastrous accident would be back in the U.S. for heart surgery immediately. A bomb test would be so controlled and automated that no one could possibly sneak on the test site, and there's no way ANYONE could get to the location in time to do anything about it; likely, it would be automated and any people would be miles and miles away.

Although it's possible the TV Hulk's origin might still be viable.

But again... the personal drama was the draw back in the 60s, and I can't imagine if it would work today, nor how.