The story
From worry + wart, apparently chosen for alliteration. Presumably related to earlier worryguts. Attested 1956, but earlier examples are found in newspapers of the 1930s and 1940s.
Etymology from Wiktionary, CC BY-SA 4.0.
worrywart
Today it means: A person who worries excessively, especially about unimportant matters. Full definition →