The name, Seymour, does not have Jewish roots.The given name is identical in origin to the surname.Seymour became a popular name among American Jews of the early to mid-20th century.
In Jewish tradition, every Jew, male and female, is given a Hebrew name, which, outside of Israel, is used mainly for ritual purposes.Quite often, a Jew's Hebrew name is given in memory of an ancestor who had the identical name, and more often than not, that ancestor lived in Europe, where Jews (depending on the country) were known by their Hebrew names only.
In America, immigrant Jews and their descendants were eager to blend into American society, so they kept their Hebrew names for ritual purposes only, and to the rest of society were known by an American name, usually of English origin.Using the Hebrew name as a model, parents gave their children English names that began with the same initial letter or sound. Many Hebrew names begin with S, and thus Jewish parents gave their sons a variety of English names that also began with S, such as Samuel, Saul, Sheldon, Sidney, Simon, Stanley, Steven, Stuart and as you have noted, Seymour. (In Germany, the same process occurred and many Jewish boys were given the very Germanic name of Siegfried.)
Today, many Jewish boys have names like Sean, Scott and Seth and other popular, or even trendy, names.At the same time, many Jewish parents are doing away with the dual naming practice and are giving their children a Hebrew name (spelled in transliteration) as both their Jewish ritual and secular, legal name.
(Many English names, such as Samuel, Saul and Simon are Hebrew in origin, but in Hebrew are pronounced differently and are transliterated in English differently. Thus, using the above three as examples: Shmuel, Shaul and Shimon.)