The story
Probably onomatopoeic. The earliest found occurrences of spelling variants (it is not established whether the first one coined the term or committed to print an earlier oral expression): * In 1923, the spelling "heeby jeebys" was used by U.S. cartoonist Billy De Beck in his comic strip Barney Google (New York American, 26 October 1923). * In 1925, the spelling "heebee-jeebees" was used in U.S. magazine Ladies' Home Journal (Vol. 42, Part 2, p. 44). * In 1926, the spelling "heebie jeebies" was used by U.S. singer Louis Armstrong for his single "Heebie Jeebies" (written by Boyd Atkins). * In 1928, the spelling "heebee-geebees" was used in U.S. article "We All Have the Heebee-Geebees" by G. M. Keller of Swift & Company (Modern School Store, Vol. 11, p. 269 via Google Books). * In 1956, the spelling "heeby jeebies" was used by U.S. singer Little Richard on the cover of his homonymous SP record (and the hyphenated spelling "heeby-jeebies" on its vinyl disc itself).
Etymology from Wiktionary, CC BY-SA 4.0.