That’s because there was essentially no actual science in the film. This one is just second to The Core in terms of gathering a great cast and spending millions on special effects and staging sequences all surrounding an utterly absurd essential plot element and then adding completely unnecessary drama on top of it.
For the o.p., no, there is nothing we can do with anything like conventional technology–not even massive thermonuclear weapons–which would make any significant impact on stellar characteristics or evolution.
The problem is that it’s not just a matter of adding more hydrogen. You can add all the hydrogen you want to the outside, but all that does is raise the pressure/temperature in the core and cause it to burn what little fuel it has even faster. Some of the very biggest stars burn up the hydrogen in their cores within hundreds of thouands of years. The fact that they only burned a tiny percentage of their total hydrogen is no consolation to the planets that were blown up in a supernova before they could even form.
Really, if the goal is to keep the star going forever, the problem is that the sun is already too big. A red dwarf star puts out a lot less energy, but it uses its fuel much more efficiently so that more hydrogen gets burned over a lifespan that would be twenty or more times as long as the sun’s.
Therefore, if we’re considering outlandish things like moving stars and planets, my recommendation would be to build a red dwarf. The smallest red dwarf stars need only about 1% of the sun’s mass… which, alas, is still something like 200 Jupiters, and still makes it seem quite easy to just move the Earth instead.
You’d have to be really close to it, obviously. And I’m not sure I’d call it normal, but as a panacea for an irrational fear that the sun might break, it does seem like the least likely to cause anxiety.
Stranger is the egghead scientist in the movie telling all the heroes how this crazy plan can’t possibly work. Fear not, he’ll get his just desserts in a tragic yet comedic death about 2/3 through the movie.