I think the short and boring answer is 8 electrons can arrange into 4 electron pairs, which gives you tetrahedral symmetry with the atom in the center and its bonds extending towards the corners of the tetrahedron. As an example, CH4 has this structure. For many atoms, 4 outer electron pairs seems to be optimal in sense that atoms still can get close enough to share electrons without bumping to each other, and the pairs can still arrange into structures called orbitals where they can get as far as away from each other as possible in a deliberate way that is described by quantum mechanics.

When atom is floating alone in space, the orbitals are all distinct and create these quantum-mechanically allowed non-overlapping structures such as s, p and d orbitals, and so forth. When other atoms enter the picture, the situation changes and the orbitals are said to hybridize, which is to say that they are no longer like that but tend to combine and the picture is now more complicated. As an example, tetrahedral symmetry is result of 2 distinct orbital shell types combining together to yield this new structure of 4 identical covalent bonds.

First group elements tend to only create one covalent bond as their outermost shell is single spherical structure that can only fit 2 electrons, and they already have one themselves. Most other elements seem to prefer 8 electrons, likely because of the sweet spot of maximizing electromagnetic attraction with electrons and protons, while also still minimizing the electromagnetic repulsion between the electrons. Then there are transitional metals which are larger in diameter and create more complicated covalent bond structures, apparently between 12 to 18 electrons, and there electrons also make use of the d orbitals which tend to be more pointy and narrow in their shape, which is a general trend with all the higher orbitals.