“Peculiar” comes from the Latin peculium, literally “property in cattle,” a meaning that lingers in “peculiar to,” meaning “belonging solely to.” Its “odd” sense arose after the term evolved to mean “distinguished, special,” describing a person or thing of great wealth or renown.
Peculium was used to describe property in general, for cattle were considered the most important form of property, and wealth was often measured in livestock. Cows were especially considered a mark of wealth and social standing because they were used to expand the scale of crop production, to haul goods and weapons in trade and war, and were also (obviously) a rich source of protein.
Owning, purchasing, and breeding cattle was considered an investment in the future, and practices involving cattle were handed down from generation to generation—meaning that cattle played a not-insignificant part in establishing early ruling classes and families.
Peculium gave rise to peculiaris, “belonging exclusively to one person,” which carried its meaning over to English in the 15th century. From there, in the 16th century, “peculiar” came to mean “distinguished or special,” suggesting someone endowed with great wealth (not necessarily in cattle) or esteem, or something else particularly renowned or remarkable.
This noteworthy sense led to what might be considered a more common use of the word today, “unusual, strange, curious,” from the 17th century—though, of course, its sense of belonging specifically/especially to someone or something remains in use today as well.