Origin of surname Underdown
Here is one of several well researched references on the origin of the family name, Underdown:
English Surnames, by Mark Anthony Lower
On pages 61-62, he discusses names that “derive from the original bearer’s residence, such as a hill, dale, or wood.” He specifically mentions Underdown as derived from “under a hill”.
On page 64, he mentions Underdown as one of several names which start with a prefex associated with a geographical feature or local situation: Bygrove, Bycliff, Underhill, Undercliff, Bywater, Bygate, Underwood. Byford, Bythesea.
There are several such references available on the Internet, all indicating that the name Underdown originated among people who lived near hills, called Downs, in the south of England, i.e., Under the Downs.
Some of the references mention the county of Kent as a notable (though not unique) source of Underdowns.
Based on early census data (e.g. UK 1901), most Underdowns originated in Kent, which has a notable geographic feature: a range of low hills called the Kent Downs or North Downs. The earliest written reference to Underdown is from Kent(Wulfinus Vnderdune, in the records of the Templars in England in the 12th Century, 1185).
Neighbouring Sussex has the South Downs and there have been Underdowns living there throughout history. There have also been a significant number of Underdowns living in Devon for many centuries. It's not clear if they originated from the Kent Underdowns or are a separate group.
Underdown and Underhill are almost equivalent in meaning but appear to have originated separately. This is my conclusion, based on the fact that the two names had quite different patterns of geographic concentration in the UK. Whereas Kent is the dominant location for Underdown, implying a connection to the Kent Downs, Underhills are found mostly elsewhere.
There have been some posts on this and other bulletin boards claiming that the surname Underdown is connected to a type of wool of the same name. No references were given, which doesn't surprise me. I have looked into this issue and cannot find any references to support this claim. In fact, the references on this type of wool point in an entirely different direction - to Kashmir, which is not anywhere near the south of England, where we Underdowns originated. Geneology is a challenging enough field without people injecting fantasy into the discussion!