This is far from a comprehensive answer, but one really interesting perspective is that of Lucretius, a Roman poet of the 1st century BC who wrote one epic poem, the De Rerum Natura ("On the Nature of Things") which aimed to provide a scientific/philosophical account of the natural world according to the Epicurean (philosophical) worldview. The Epicureans were atomists, meaning that they, like us, believed that tiny, indivisible atoms were the building blocks of all matter.

According to Lucretius, both men and women produced a kind of sperm (semen/semina) during intercourse. The two lots of sperm (or possibly the two lovers? Battle was a standard metaphor for erotic activity in Classical literature) then engaged in a sort of battle. Depending on which partner/partner's sperm was the strongest (vis), the child would look like one or the other or both. (Lucretius, DRN 4.1209-17).

As to why characteristics sometimes "skip" a generation, Lucretius argues that this is because "the parents often conceal in their bodies many first-beginnings [atoms] mingled in many ways, which fathers hand on to fathers received from their stock (DRN 4.1218-26, translated W. H. D. Rouse). This is somewhat analogous to our modern concept of recessive vs. dominant genetics. Part of the Epicurean theory of matter included the "seeds" theory, which essentially explained why (for example) giraffes mating always produce giraffes rather than trees or rocks or camels, barley seeds will always produce a barley crop, etc, and Lucretius uses the same vocabulary for "seeds" as for "atoms". This "seeds" theory seems to be what lies behind this otherwise confusing passage, which explains that the reason why some traits are latent in the parents but manifest in the offspring is that these traits were present at the atomic (or "seed") level in the parents.

The idea that both females and males produced sperm seems to have been quite common in antiquity. This article by Pieter Willem Van Der Horst goes into detail on both Greco-Roman and ancient Jewish theories of reproduction, chief among which are the "sperm-battle" idea (or at least the idea that both sexes produce sperm), and the idea that the man's contribution to the matter of the developing foetus was, of course, semen, while the female's was menstrual blood. More specifically the article is aimed at showing that the wording of Hebrews 11.11 in the original Greek (Sarah's miraculous conception of Isaac from the Old Testament) reflects the contemporary (1st-century AD) understanding that women also produced "seed".

Hope this helps!