Beside names which reflect place names (here) and occupations (here), a third very common source of surnames are partronymics, names which originally indicated the father of the name's bearer. These types of surnames often incorporate the word for son (or in some cases daughter) and are common in many languages. In some languages, like Icelandic, patronymics are still productive. For example Icelandic politician Björn Bjarnsson is the son of former Primer Minister Bjarni Benediktsson. The son's last name is determined by father's first name. In many other languages, including English, the patronymics have been fossilized and are passed like any surname from a father to his children.
In English, there are two basic types of surnames derived from patronymics, those ending in -s (Williams, Adams, Roberts, Rogers) and those ending in -son (Williamson, Robertson, Johnson, Jackson). The forms with -son are most common in areas where there was earlier a strong Norse or Scandinavian presence, such as Northeast England and the Scottish Lowlands. In the United States, names ending in -son are often associated with the Scotch-Irish. Many of the Scotch-Irish lived in these parts of Great Britain before they settled in Northern Ireland and eventually moved onto the United States where they formed a large part of the early frontier states, like Tennessee. My own Scotch-Irish Simpson ancestors made their way from North Carolina, settling in Cannon County, Tennessee for the first half of the nineteenth century, then briefly in Lawrence County, Missouri, then moving to Lawrence County, Arkansas, and finally to Washington State.
The name Simpson is clearly patronymic in origin, but the meaning of the parts is not obvious. First, the given name included in this surname is an unfamiliar shortened form of the name Simon. Like surnames derived from place names, surnames derived from surnames may retain archaic forms that reflect an earlier stage of a language. The name is literally "Sim's son".
Whence the /p/?
This brings us to the second problem, where the /p/ in the name comes from. This is where the linguistics comes in. The /p/ is the result of a common process that occurs when certain kinds of sounds come together. In the case of Simpson, there is an /m/ followed by an /s/. The sound /m/ is called a nasal sound. This means that the sound is produced by lowering the soft palate and allowing the air to flow through the nasal cavity. If you make an /m/ or /n/ sound you can feel the air coming out your nose. The /s/ is an oral sound, meaning the air is flowing through the mouth. The /p/ emerges because of the timing of these two sounds. Notice that the /m/ and /p/ are both made by pressing the lips together. First, the lips are closed and the soft palate is lowered to make an /m/ sound. Then, the soft palate is raised closing off the nasal cavity, the pressure behind the lips builds up, the lips are opened creating a /p/ sound and then the tongue goes into position allowing enough air to pass through to make an /s/ sound. Thus, we have a new sound creeping into many words.
The same processes is also found in many other surnames, such as Thompson "Thom's son", Sampson "Sam's son", Hampson, Lampson, Hampton, Hampstead, etc. Sounds like /p/ which are not original may not occur in every spelling of a name, especially in older documents and inscriptions. There can be a lot of variation in spelling, including with and without the /p/. This is important especially when using the Soundex system, because Simpson (S512) and Simson (S525) will have different codes.