A reference that is old, has no bibtex entry, and probably not well-known throughout the readership. How to cite this?

For cases like this, unless you want to give a historical reference, there's usually no need to cite the primary source, you can just cite your favourite circuit theory book.

Anyway, if you really wish, you can certainly cite the original paper too, which can be found, e.g., here.

For what concerns the added question on where to draw the line, you can have a look at the following question, and the answers therein: How generous should I be with citations?

I am writing an article for an audience that mostly consists of computer scientists. A specific part of the subject is motivated by Kirchhoff's laws. I intend to mention this fact and reference the rules. But that is easier said then done. Apparently, Kirchhoff predates referencing.

Your statements make no sense. It's like saying that you want to use Pythagoras's theorem "but that is easier said than done. Apparently, Pythagoras predates the English language." It is trivial to cite any paper you wish: just write the necessarily BibTeX.

The fact that no references appear in Kirchhoff's paper is completely irrelevant to whether or not you should use references. You are writing today, not in the 1840s, and today's standards apply to you.

How to reference a source that is so old, it has no bibtex entry, but is also probably not well-known throughout the readership?

You write you own BibTeX entry! BibTeX is just a language, like LaTeX.

However, it isn't necessary to give citations for such basic material as Kirchhoff's laws. They're part of standard high school education, so you can assume that everybody knows them. And, even if somebody doesn't know them, the phrase "Kirchhoff's laws" is specific enough that they can Google it and get the answer.

For well-known basic physics, it is not necessary to cite anything. But if you do include a citation, you should cite the original source (regardless of its age), and to be helpful to the reader, also cite one or two recent textbooks.

If you only cite a secondary source (“your favourite circuit theory book”), then you are misleading the readers, making it hard for them to verify what you are saying, and quite possibly propagating information that is incomplete, misleading, or inaccurate. Ole Bjørn Rekdal has written several excellent articles about these issues. Here are parts of two of the abstracts:

• From Monuments to Academic Carelessness (2014):

  [Katherine Frost Bruner's] collection of advice to writing scholars has been widely quoted ... The most frequently quoted message in Bruner’s article deals with the importance of making sure that references in academic texts are complete and accurate. Exploring the citation history of this particular message reveals an ironic point: the great majority of those who have quoted Bruner’s words on reference accuracy have not done so accurately.

• From Academic urban legends (2014):

  Many of the messages presented in respectable scientific publications are, in fact, based on various forms of rumors, [because] authors have lazily, sloppily, or fraudulently employed sources, and peer reviewers and editors have not discovered these weaknesses in the manuscripts during evaluation.