The equivalent in English for 本科教育 is undergraduate education, but in China this phrase may imply it is a university course that lasts four years, different from 专科教育, which is the type of college education spanning over two or three years. The word 本 means origin or the true self or nature of something or somebody, but when put in this context, makes the phrase 本科 a little confusing.
I think you have actually answered the question when you say -- "The word 本 means origin...."
Because "origin" is defined as -- "The place where something begins", "An event that is a beginning; a first part or stage of subsequent events"
So, the 本 in 本科教育 means the "beginning", the "start", the "first stage" of one's tertiary education as an undergraduate. Therefore 本科教育 means undergraduate education.
Wonder what the word "under" is doing in "undergraduate"?
That is why Japan is called 日本, because it is the "start", the "beginning" or if you like the "origin" of a new Earth day as the sun "日" "starts", "begins" shining on that country "first"
So too you have 基本, meaning "basic".
Further Answers from Comments:-
As you know the word 本 has multifarious meanings. I confined my answer only to the question concentrating on the meaning of "origin".
The 本, however, in 本职工作 means "self", as you've said "...true self or nature of something or somebody" So you have, 本人, (myself), and 本性, (own / self personality)
Thus 本职工作 means, "self-employed", (本 = self; 职 = occupation / job)
The word 专 too has many meanings, like all words in any language.
But in 专职工作, the 专职, (expert / specialized occupation / job), has the meaning of some sort of "technical specialization", like an engineer, etc.
Related terms are, 专门, a person who does something or work which he has special skills in. Same as 专家, an expert.
本科教育 vs 专科教育?
By now you should have no problem with these terms, and or their comparative meanings :)
In many areas of human endeavor, there is nothing better or more satisfying than self-education.