Why is hydrogen the most abundant element in the Universe?
The short answer is that protons (hydrogen nuclei) are produced abundantly in the early universe, but only a small fraction of these are able to engage in nuclear reactions leading to heavier elements, either during primordial nucleosynthesis or later inside stars.
In the early phases of the big bang only the constituent parts of nucleons (quarks and anti-quarks) plus leptons (e.g. electrons, positrons) and light existed. As the universe expanded and cooled, quarks were able to combine and form the basic building blocks of nuclei - the neutrons and protons. A proton is of course a hydrogen nucleus; any heavier elements needed to be built by fusing together protons.
Protons are positively charged and repel each other strongly. In order to fuse protons and make helium requires high energies/temperatures and the intermediate stage of forming deuterons - a proton plus neutron bound pair.
The reason that most protons do not fuse in this way is threefold.
The small excess mass of the neutron compared to the proton means that the equilibrium reactions that produce nucleons end up producing about six times as many protons as neutrons.
Free neutrons then decay into protons on timescales of minutes, which increases this ratio further to about seven.
The deuteron is weakly bound, so there is only a limited window of time between when it is too hot to remain bound (prior to 10s after the big bang) and when it is too cool to get deuterons to fuse to become helium (at about 10 minutes after the big bang).
Of these things, by far the most important factor is #1 and this is fundamentally the answer to your question. It is because the neutron is more massive than the proton. All heavier nuclei must contain neutrons because the Coulomb repulsion between two or more protons is too strong to form stable nuclei without them. By the end of big bang nucleosynthesis, essentially all of the available free neutrons end up in helium nuclei (with traces of deuterons and lithium nuclei) and thus the H/He ratio ends up at 12 protons for every He nucleus (2 protons, 2 neutrons).
Is this quantitatively precise? Well, my figure of 12 protons for every He nucleus, or 75% by mass, is a bit handwaving. A detailed model must take into account in a more accurate way the temperature evolution of the universe, the various (more minor) nucleosynthetic pathways, photodisintegration and so-on, but yes, the models do provide a very precise prediction of the H/He (and the minor species) at the end of cosmological nucleosynthesis. There are small uncertainties in the neutron lifetime, the various reaction cross-sections and so-on, but the most important uncertainty is that there is an important "free parameter" - the baryon to photon ratio that must be fixed. This can be constrained by demanding that one consistent value of this number can explain all the measured primordial abundance ratios (He/H, Li/H, D/H, $^3$He/H) or it can be found from other cosmological measurements (such as from the cosmic microwave background).
Current levels of uncertainty on the mass fraction of hydrogen produced in the (standard) big bang are significantly smaller than 1% (e.g. Peimbert 2008). Coc et al. (2013) use the Planck constraints on the baryon to photon ratio and estimated a standard big-bang He/H mass ratio of $0.2463\pm 0.0003$. One could consider additions to this standard model - e.g. change the number of neutrino families, have decaying dark matter particles in the early universe and so-on, but it seems hard to change the mass fraction of hydrogen by more than 1% without upsetting the concordance with other observations.
A final point to make is that very little of this cosmological material has so far found its way into stars (perhaps 10%), and of that, much of it is still in the same (low-mass) stars that were formed. The amount of "processed" material made of heavier elements, fused from hydrogen in stars, that has enriched the cosmological material is therefore comparatively small - of order 1%. So the predominance of hydrogen from the big-bang has not changed much.
This latter property can be used to test the whole model. By looking at the He/H ratio as we go "back in time" we can see if the primordial ratio matches that predicted by the big-bang. In practice this can be done by estimating the He/H ratio in the oldest stars or by estimating He/H in the interstellar medium of the most metal-poor galaxies. These measurements are more uncertain than the predictions above, but are in reasonable agreement with them. An example would be Izotov & Thuan (2010), who estimate a primordial He/H mass ratio of $0.2565 \pm 0.005$ from metal-poor galaxies - about two (small) error bars higher than the prediction above.
Between about t=$10^{-12}$ and t=$10^{-6}$ seconds, the universe was filled with a quark-gluon plasma. Temperatures were too high for mesons and baryons (like protons and neutrons) to remain bound, should they briefly form.
Then, between about t=$10^{-6}$ and 1 second, during the "Hadron epoch," the temperature was low enough that quarks that stuck together could remain together. Statistically, the simplest stable configurations were protons and neutrons, so that is how the majority of quarks ended up bound.
During the next 200 seconds, temperatures were still high enough for nuclear reactions, so about one-quarter of baryons (neutrons and protons) were bound together into helium (and traces of other elements).
During the next hours, any remaining neutrons decayed into protons and electrons (and neutrinos).
And that's roughly where we're at. Solo protons are H$^+$ (hydrogen ions) and those that combined with electrons are H (hydrogen atoms).
Cosmological nucleosynthesis furnishes a remarkably accurate accounting of the amount of both hydrogen and helium in the observable universe. The satisfactory correspondence between the amounts observed and the amounts predicted by the standard big bang model is one of the main reasons for the general acceptance of the validity of the big bang model.
Weinberg's book "The First Three Minutes" provides an excellent description of these processes and is highly recommended.