Possibly, the name Jebediah derives from the name Jedediah. If so, then what phonological phenomenon is this an example of, and what are other examples of it?
This could be an example of dissimilation. One prevailing theory about the mechanism for this kind of phenomenon is the Obligatory Contour Principle (OCP). Similar (output) patterns are well-attested in Arabic, for example.
Here is a quotation from Frisch, Pierrehumbert, & Broe (2004), referencing the history of this observation: "Greenberg first observes that there are no roots that repeat the same consonant in first and second position (e.g., ∗dadam, though McCarthy (1994) reports one verb of this form). Many verbs are found with identical consonants in the second and third positions of the root. Examples include madad ‘stretch’ and farar ‘flee’."
You may notice right away that the Arabic pattern is different from your example (the constraint applies to first and second consonant position, not second and third).
It can also be a contamination from the biblical name Zebadiah.
Zebadiah mixed up with Jedediah gives Jebediah.
P.S. (This is an answer to the question in comment: Is it a joke?) It is partly an elaborated joke indeed. The name Jebediah was popularised by the TV show Simpsons. But Jebediah Smith (as founder of Springfield) is an anachronism.
The name is in use in the USA, as the SSA data show, since 1973.
This trend was caused by the figure Jebediah Nightlinger in the film The Cowboys (1972)