If it were possible for us to create a star or a new sun, how could it be done, or how would it have to be done?
Answering a question like this requires a shift in thinking from normal physics & engineering to something akin to the futuristic thinking accompanying the concept of a Dyson Sphere: https://en.wikipedia.org/wiki/Dyson_sphere With this in mind it seems that the only possibility for humankind to "create" a star in the foreseeable future would be through the "butterfly" effect. By that I mean introducing a small perturbation into a distant cloud of gas and dust that could get magnified by extreme sensitivity to initial conditions so that the resulting "storm" would be the formation of a new star. The perturbation could be delivered by a small space probe sent into an appropriately chosen cloud of gas in our galaxy: a cloud that already possessed the potential for star formation. Do we know how to do it now? No. Could we learn by intensive study? Yes. The possibility for funding such a project might improve in a few billion years as the red giant phase of our sun draws near (if man kind's propensity for science continues).
So what are the conditions required for the butterfly effect to manifest itself. The first requirement is a nonlinear system. The second is that the system exhibit dissipation (friction or viscosity). For fluid systems described by the Navier Stokes equations, both of these conditions are present. A final requirement is that the system be subject to periodic or aperiodic driving forces. Two gas clouds in orbit about each other and deforming via tidal forces fulfills this last requirement.
The ALMA observatory has recently produced a striking image of star formation in just such a scenario: http://www.almaobservatory.org/press-room/press-releases/723-dynamical-star-forming-gas-interaction-witnessed-by-alma- Following this link one see a newly formed star (a protostar) between two gas clouds. Below the astronomical image is a computer simulation of the dynamic interaction for two such fluids. In the region of overlap between the fluids the excessively turbulent rotating flow greatly resembles a hurricane or tornado. The implication is that such "stormy" conditions are indeed the cradles of star formation. This casts doubt upon previous views of star formation that supposed the process to proceed slowly via gravitational contraction of gently moving gas clouds. These stormy situations are where the colorful "butterfly effect" terminology originated.
So what are the possible "butterflies" that man kind could use to influence such a potentially star forming region and how could we ever know that we were in fact responsible for birthing a new star? Suppose we found a situation where the two gas clouds were just beginning to overlap and we were able to arrange for our probe to arrive on the scene (slowing down to just the right speed) at just this moment. We might "seed" turbulence into the clouds by releasing atomic oxygen from our rapidly spinning probe. The oxygen would react with hydrogen already present in the cloud to form a small vortex of water molecules and the nonlinear dynamics of storm formation could take it from there.
A much more likely scenario (although still exceedingly difficult with today's technology) would be for us to arrange for a probe to arrive after some turbulent motion had begun but before significant gravitational nucleation had started. This time our probe could release a strongly magnetized hunk of iron and seed the accretion process as iron dust (already present in the cloud from previous supernovas in the vicinity) was attracted. This magnetic attraction could significantly jump start the nucleation process since the electromagnetic force is many orders of magnitude stronger than the gravitational attraction.
The question of "how would we ever know" that we were responsible is much more difficult to answer. Stars are formed all the time, after all, without our assistance. The answer to that will have to come from future advances in our understanding of nonlinear dynamics perhaps learned from interactions with a civilization that already inhabits a Dyson Sphere.
Step 1: Locate hydrogen source.
Step 2: Create a massive shock wave which will compress the gas in some parts and rarefy it in others. This gives you a mass gradient.
Step 3: Wait.
Step 4: Hopefully in a few million years time, the concentration of hydrogen is critical enough that it starts to sustain thermonuclear reactions.
There is no way to "ignite" a gas giant: even if you could temporarily start fusion in it's core, any artificially-generated reaction like this would be self-limiting: the energy it creates causes the proto-star to expand a little bit, and then nothing else happens.
If we had infinite time, we could always create a star by just collecting an absurd amount of matter and bringing it all to the same place, but it's hard to imagine this ever being a worthwhile endeavor.
This is my guess but seems possible. If there was very cold gas giant (few times grater than Jupiter) than one could try to "ignite" it - start nuclear fusion. It has to be very cold so that reaction would not be started automatically. The idea is to drop a bomb (hydrogen for example) to the core of the planet. Bomb has to withstand extreme preasure while falling into the core but shield has to be weak enough to break due to presure from the inside of the bomb after explosion.
Have you ever read Arthur C Clarke's book called "2010"
Intelligent aliens increased the mass of Jupiter until it reached a critical point and collapsed into a small star. The reason was to create a small solar system with the moons of Jupiter increasing the chance of life on Europa.