Godzilla is already a household name. However, he's not popular enough to rival franchises like Harry Potter, Star Wars, and Transformers. It's not because audiences wouldn't pay money to see giant monsters fight. GODZILLA '98 was so close to being a huge success because of that very reason. It only failed because it didn't meet expectations.
So there's a desire for it. That's why Legendary Pictures is gearing up to (hopefully) give us what we've wanted been clamoring for for years. The best way is to make Godzilla popular is by making him a monster at the box office (no pun intended). This can be accomplished in two ways:
1) Take the big budget, Hollywood action/adventure route. Give us exciting visuals, sequences, and battles. Entertain the hell outta us. Make sure the human characters aren't made of suck. They have to be strong enough to carry the film when Godzilla and his kaiju brethren aren't present. Give them something to do. Hell, make us care about whether or not they die. Is the main character trying to find his or her lost child in a city gone mad? Have the main characters been poisoned by the radiation and are trying to find a cure? Are we following a group of soldiers instructed to take Godzilla down at all costs? Subplots like that are fun and exciting to get into.
Finally, make sure it's a big universe we're visiting. Keep it ambiguous. More importantly, keep the mysteries mysterious and give us answers that only lead to new questions. If your audience wants to know why these monsters exist, you're on the right track! How you know you have a franchise is by making your audience feel like they already live there. Halo is very successful franchise because it has many video games, books, graphic novels, and films.
So give us a fun, exciting, imaginative universe, and in return we'll keep coming back for more.
2) Go low key. There are different ways of doing this. We could be introduced to several different people, see how they live, and then watch--either in horror or in amusement--how their lives change forever. Maybe it's more like a natural disaster film. Godzilla is more like an earthquake, he comes and goes, and we see how the aftermath plays out. It could become more of a human survival picture. Basically, it focuses less on monster mayhem, and more on on the plot/characters/symbolism.
This could work. I'm sure it'd make for a damn good film. From a critical perspective, it'd be a nice change. Imagine coming out of a giant monster movie going, "Man, that film was deep, it totally made me think about how dedicated the human spirit is, how wonderful it is to be alive, and how [nuclear weapons/natural disasters/human greed or folly or pride] really suck!"
#1 has a better shot at making Godzilla popular (i.e. like Transformers, Pirates of the Caribbean, Iron Man).
#2 has a better shot at bringing Godzilla critical reception, winning awards, and having audiences take him seriously.
Of course, I'd love to have both, which is possible. In fact, you can have both if you have a good script and direction. I'd love to have a blockbuster, mega-box-office hit, critically received Godzilla film. Having both usually doesn't happen though. One can hope.