Since this is a plot point that has been constantly debated throughout the entire run of the Stargate franchise, here's something I would change:
Make the Stargate public knowledge from the very beginning, and actually allow the world to react to the news instead of keeping it a secret the whole time.
That's the kind of thinking I'd want to see from the people I hire to reboot it - don't get hung up on preconceived notions about how a show must be this way or that way. Be daring or don't bother. Maybe that's not the right approach but someone needs to give it a good think-through before coming to that conclusion.
Well, a Stargate reboot does have the really unusual out of screwing continuity all to heck and calling the program "Wormhole Extreme!"

"Of course it's totally wrong. It has to be because this show is a cover for the real show, unless it was the other show that was the real cover, and this one is..."
It might be easier to think outside the franchise and pull in vaguely similar premises from science fiction.
Here are a few ideas:
1) More advanced and diverse human enemies.
The Go'uld kept humans as primitive laborers, but perhaps it would be more interesting if the alien races had been using us because we're adaptable, trainable, and good with technology (as opposed to pushing plows and stabbing people), perhaps much more so than the aliens.
Then instead of constantly encountering medieval villages and Egyptian settlements, we're encountering technically advanced humans who could believe all sorts of crazy things, making them far more dangerous and threatening than the Jaffa ever were.
Stargate used elements of advanced humans many times, but usually those plots were written as a way for us to get some non-Go'uld technology to fight the Go'uld, or to provide a doomed love-interest for Carter. Humans were never a significant adversary on their own till the show ran out of Go'uld and came up with the post-Go'uld Lucian Alliance, which was never very well developed. Basically, all the humans we fought seem to be either still enslaved, free but primitive, or using Go'uld tech. The humans who had advanced were allies, if often irritating ones.
What if there were very advanced humans out there, coming from a variety of alien "owners" (so their technology and cultures would vary more widely), and upon discovering us, they frequently decide to reclaim their Earth homeworld or destroy it before someone else can claim it, or naturally assume that they should be running it?
2) Greater diversity of conflicts
What if the humans often fight bitterly amongst themselves, perhaps in concert with aliens or dragged into alien wars, providing any number of story arcs as needed?
That would allow shades of Babylon 5, including occassionally losing the Earth to hostile forces through direct action, intrigue, and misguided trust of Greeks bearing gifts, while retaining the exploratory character of Stargate (along with fun stand-alones) by not clinging to a single arc.
3) Greater diversity of rules (what works where, and why)
You could make it sort of a meta-science fiction show, where we might encounter equivalents of Colonials fighting Cylons, the Federation fighting Romulans, rebels fighting the Empire, not locking things into a particular type of ship or set of rules. This could prevent the characters from reaching the state where they have everything figured out and all the tech they need, which afflicted SG-1 in the later seasons.
4) Divide the galaxy into zones.
One way to do this is to have the galaxy divided into zones where the rules of physics are slightly different, like Vernor Vinge's "Fire Upon the Deep" to explain why ships and technology don't seem to work universally, why ships in some areas "jump", others warp, some use fusion drives and missiles, and some don't. The nature of the different zones would keep one type of technology and culture from becoming dominant everywhere, and provide a reason for episodes that only work with certain tech (this plot needs a warp core breach, that plot needs a starfighter to jump into hyperspace).
This would have to be spelled out explicitly very early in the show so viewers don't think the writers lost their marbles and are making things up on the fly without any regard to continuity, and would have to be brought up periodically as a reminder, perhaps with plots concerning aliens trying to obtain weapons and drive systems that would let them expand into a neighboring zone.
The one constant would have to be the gate system.
The reboot:
Perhaps the best way to reboot into this would be the discovery that the equivalent of the Ancient weapons platform on Tokara was used to rewrite the rules of physics across local domains, which was done to prevent any one of a number of threats from taking over the entirety of the galaxy just by possessing an edge in just one propulsion or weapon technology. It would be the same as preventing the possibility of a single predator from becoming dominant over a planet by dividing the planet into wildly diverse ecosystems, knowing that no animal would work well in swamps, oceans, deserts, forests, ice flows, and grasslands.
For continuity, the SGC could notice that the Go'uld, Replicators, and other threats were only operating in certain areas of the galaxy, travelling above or below the spiral arms when they crossed certain regions, which wasn't very obvious when the Tok'ra map display only showed the galaxy from the top. Then we notice that the Gate system doesn't seem to extend into any of these areas, either.
As we send ships to try and explore the neglected zones, their drives cut out, and we realize that the physics are changing, and different. Our ships have to back out slowly on reaction thrusters. Probes show that even F=ma and momentum breaks down a little further in, and the speed of light drops dramatically, so the only way across would take thousands of years of low-speed coasting.
We consult the Asgard database and they never found a solution, having explored almost every conceivable propulsion system. The Go'uld never took things further than that, and both they and the Asgard assumed the whole region would be similar, and assumed that the Ancients never put gates there because the Ancients couldn't get in there, either.
But we discover that the problem is one of the borderlands between zones, where one set of physical laws is being replaced by another set, and that both sublight and FTL propulsion is still possible within the zones, away from the borders, and that there are gate systems within the zones, too. We figure out how to use the Ancients' equivalent of dialing a country code with a password to get between zones and explore what's inside, realizing that the Ancients split the gate system into seperate sections to reduce the chance of a single species takeover, the same reason they'd created the zones themselves. Inside the zones we might discover - a whole new series! ...Stargate Ecosystem...
And then the whole thing is rebooted.
