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The word "develop" has applied to projects longer than we've had computers. Merriam-Webster has the definition "to create or produce especially by deliberate effort over time". Computer programs require this kind of work, so the word fit.
Google Ngrams shows a huge spike in the use of the term "research and development" around WWII. I'm having trouble linking it, but you can easily reproduce the search.
The earliest example I found of the term applied to computer programming is a chapter called "The Program Development Cycle" in Spencer's "Introduction to Information Processing" of 1974. The chapter is full of italicized key terms (instructions, program, flowchart), but "development" is not given this treatment. It wasn't yet a term of art, just the word for a process of creation.
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According to the Oxford English Dictionary, the word "developer" (in any context) dates back as far back as 1772:
A late excellent developer of the human heart...
Letters Several Eminent Persons Deceased
There have been a lot of other things and people called "developers" since then. In particular, it has been used to describe "[a] person, organization, etc., that develops a new product or technology" (OED) since 1905:
[O]ur chairman, the reinventor and the leading developer of the spectroheliograph...
Popular Science
As for in the software sense, the earliest example given in the OED is from 1961:
Roderick D. McIver, program developer for the GE computer group will be host.
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