In addition to G. Smith's answer, it's worth noting the importance of the neutron-proton ratio.
Two key nucleon reactions before Big Bang Nucleosynthesis (BBN) commences are:
- a neutron interacts with a positron to create a proton plus neutrino (and vice versa), and
- a neutron interacts with a neutrino to create a proton plus electron (and vice versa).
Initially the temperature was high enough for all these reactions to take place, maintaining an equilibrium of protons and neutrons. However, as the temperature rapidly dropped, the neutron-proton inter-conversion rate per nucleon fell faster than the Hubble expansion rate, favouring protons ahead of neutrons, until the temperature had dropped to about 0.7 MeV (the "freeze-out", about one second after the BB) at which point the N:P ratio had fallen to 1:6.
BBN starts at this point, and the first multi-nucleon element formed is the simplest: deuterium (one proton plus one neutron). However, the temperature was still too high for deuterium to survive, as the energy of some photons was higher than deuterium's binding energy and the element would quickly photodissociate. This period is called the "deuterium bottleneck"; once the temperature had dropped to about o.1 MeV there was a sudden burst of deuterium formation.
As helium-4 has the highest binding energy per nucleon among the lighter elements, almost all the deuterium ended up as $^4$He. However, not all neutrons managed to get to "safety" within an atomic nucleus: free neutrons are unstable and decay with a half-life of 611 seconds, and some of these decayed into protons before they could fuse with a proton. As a result, the final N:P ratio ended up at 1:7.
This ratio tells us that for every 14 protons there were two neutrons. Since on average those two neutrons end up in a $^4$He nucleus with two protons, this leaves 12 protons without a partner. A proton on its own is a hydrogen ion, so we can predict that once the temperature had dropped too low for further nucleosynthesis (at around 30KeV, or about 20 minutes after the BB), there were roughly 12 hydrogen atoms for each atom of $^4$He.
[There were also trace amounts of other nucleosynthesis residues: about 0.01% of deuterium and $^3$He, and one part in 10 billion of lithium-7; and also some tiny amounts of the unstable isotopes $^3$H (tritium) and beryllium-7, both of which soon decayed.]
So, this is what our knowledge of physics predicts: by the end of Big Bang nucleosynthesis, on average 12 out of 13 atoms will be hydrogen (92%) and the remaining atom will be helium (8%); or by mass, with a helium atom four times heavier than a hydrogen atom, it's 4:12 or 4/16ths or 25% helium, and 75% hydrogen. And this is, indeed, exactly what we now find.
For a more technical review of BBN predictions, including pre-BBN processes and N:P ratios, see (for example) Primordial Nucleosynthesis in the Precision Cosmology Era (Gary Steigman, 2007).