This is unfortunately not something we can really know because we have very little evidence about what family structures were actually like. The archaeological evidence doesn't give us much of a hint, so the best we can do is make analogies with more recent groups that have been archaeologically, ethnographically, or historically recorded.

That said, all the evidence we have indicates that there is a huge range of diversity in family organization among hunter-gatherer groups, both at present and in the past. This does suggest to us that we likely can't make any blanket generalizations about pre-agricultural societies either.

However, in terms of monogamy and polygamy, polygamy does tend to be common among state societies as a marker of status or prestige (i.e. important people with multiple husbands/wives). That doesn't mean there can't be any polygamous hunter-gatherer societies prior to agriculture, but it may not be as common as in sedentary, agricultural societies.

I should stress also that the adoption of agriculture was a gradual process across the planet, and still incomplete at present since there are modern hunter-gatherer societies. One of the criticisms of using these modern groups as analogies for pre-agricultural societies is precisely that these modern groups are many thousands of years removed from paleolithic and early neolithic hunter-gatherer societies. Additionally, these modern hunter-gatherer societies live in a world dominated by agriculture which surely impacts how they organize themselves and other characteristics of their society.

If you are interested generally in how people live as foragers (i.e. hunter-gatherers), the book The Foraging Spectrum by Robert L. Kelly is a really great place to start reading. This book really emphasizes the incredible diversity of hunter-gatherer societies. It is this same diversity, along with a lack of direct evidence, that means I can't give you a satisfactory answer other than "it probably differed a lot from society to society".

This isn’t really a question that can be answered by history (because history only goes back around 5,000 years) or anthropology (because archaeological remains don’t reveal complex behavior such as monogamy or polygamy).

But there is a source of objective evidence which can cast some light on the problem, and that is genetics. The human population genome – differences between DNA sequences of individuals on a population level – encodes population histories. Of course, the information is not very granular, representing lineages rather than individuals, and it requires a lot of statistics to detect.

Since there are parts of our DNA which are uniquely inherited from the mother or father, (the mtDNA inherited solely from the mother and the Y-chromosome inherited solely from the father), it’s possible to differentiate and compare the maternal and paternal ancestries of humans and their lineages. One finding from such studies is that the effective population size of our maternal ancestors was larger than for our paternal ancestors. This is not only true today; it was also true at every point in history and on every continent ever since there have been humans.

The “effective population size” is a measure of variation in a population and represents the number of lineages that exist. More maternal lineages than paternal lineages implies that more women got to reproduce than men.

Several explanations have been offered for this observation, but the two with the most evidence are selective migration and polygyny. Selective migration means one sex migrated more than the other, in this case, women. This could easily result from widespread patrilocalilty, meaning women move to their husband’s home after marriage.

But recent work casts doubt on this. The reason is that patrilocality produces what we call a diffusive pattern of migrations, meaning they aren’t very far. Before the modern days of trains and automobiles, most people married within 5 miles of where they lived. So even if women migrated more, they didn’t migrate very far very quickly. This is different from long distance migrations, where entire populations pick up their belongings and settle down somewhere far from where they started. Such long distance migrations typically involve both sexes equally, or more males than females.

Recent genetic evidence shows that local migrations aren’t likely to produce the differences in effective population sizes that we observe between males and females. This has strengthened the position that polygyny is the likelier explanation. We have more female ancestors than male ancestors because a relatively small number of males monopolized more women. Most women were able to reproduce. But many men did not get the chance to reproduce, because some men impregnated more than one woman.

Here’s one of the more recent papers that goes over this:

Lippold, S., Xu, H., Ko, A., Li, M., Renaud, G., Butthof, A., … Stoneking, M. (2014). Human paternal and maternal demographic histories: insights from high-resolution Y chromosome and mtDNA sequences. Investigative Genetics, 5, 13.

A short summary of their findings is:

• This imbalance between male and female reproduction goes back tens of thousands of years, existing prior to the Out-of-Africa event around 75,000 years ago.

• It exists at every demographic split – Australia at 60,000 years, Europe at 49,000 years, the Americas at 15,000 years, etc. But its magnitude shows geographical variation, being high in Europe and the Middle East, relatively low in the Americas.

• The magnitude of this effect has gone up recently, around 15,000 – 20,000 years ago. Some people equate this with the development of agriculture, saying that agricultural societies allowed the accumulation of wealth and therefore became more lopsided, enabling some men to hoard more resources and gain access to more women, whereas hunter-gatherer societies were egalitarian. However, the timing better corresponds to the rise in population densities following the end of the last ice age, not agriculture.

Of course, it may be that there is some other explanation for these genetics, not polygyny. We can’t be very certain until there is more data to correlate effective population size histories with other events. We’ll have to wait a few more years for more people to be sequenced, but so far it seems a fairly strong hypothesis.