Snookums is a rather silly term of endearment; it is often used jokingly or with sarcasm. It may be used seriously, but even then it is associated with baby talk and particularly sappy sweet-talk. It also has a strong association with pampered, yappy lap-dogs1. In an attempt to be even more cutesy, it is sometimes spelled 'schnookums'.
Snookums is a long-standing diminutive of Snooks, an old-fashioned name for a generic person2; much as we use John Doe today; this person is probably named after the Old English snoc, meaning a projecting point of land;3 landmark-related-names are a common theme amongst hypothetical 17th century folk, as evidenced by John Nokes and Tom Styles.
Snookums first appeared in print/film in 1919, when the movie Mules and Mortgages featured a monkey named Snookums; the monkey was so popular that she later starred in her own series of two-reel comedies. The popularity of the name snookums was renewed again when the artist behind the comic strip Bringing Up Father, George McManus, introduced a new topper strip4 called Snookums. The strip featured a bratty young boy named snookums, who while not particularly endearing, reinforced the silliness of the name.
Footnotes:
1. The OED specifies that it is commonly used in reference to lap-dogs and children; I personally have not heard it used much to refer to children.
2. Snook was used as generic name as far back as the Domesday Book, which was completed in 1086.
3. It has been suggested that in this sense, snoc may have been used to refer to a person with a large nose. Clockmaker reports that "in Swedish, »snok« (long O, like the others) is a slangish word for »nose«. Although I can't recall it ever being used with a specific connotation of largeness, it's mostly used about large noses. »Honker« might be a good analogy."
4. A topper strip is a smaller strip that appeared above the main strip, which back in those days could take up most of the comic page. Snookums, even in this short form, still had as many panels as a modern day Sunday comic strip. It was also a tie-in to one of George McManus' earlier popular strips, The Newlyweds; Snookums was The Newlyweds' son, and was an early model for Calvin.