How do I find the original paper of each famous theorem? Lately, I writing some essays whose topics are mathematics-heavy. Even though they are not research papers and will never be published, I just want to give proper references to each famous theorem/ideas.
However, finding the original source of each theorem proves to be a much more difficult task than I thought. This brings me to my question

How, in general, do I find the original papers of each famous theorem?

Specifically, how do I find Caratheodory's paper on the extension theory that now bare his name?
 A: As you observe, in many cases the most celebrated results are viewed as being so widely known and diffuse that no reference is given. Yes, a bit ironic.
A way to try to circumvent that is to look at as-old-as-possible textbooks/monographs, from times within few decades of the developments you'd want to trace back. Things would seem different to those people... For example, the Whittaker-and-Watson "Modern Analysis" will give (dangit-awkwardly-footnoted-buried...) references to many things that were new then, but not now... 
Jesper L-umlaut-utzen's 1984 essay on "Sturm and Liouville's..." gives many original refs. 
There is an AMS-published volume "History of Analysis..." which has many original refs.
The quasi-encyclopedic two volumes edited by I. Grattan-Guiness (sp?) are marvelous, with nearly-infinitely-many original references.
(And, if you have the time/energy to double-check, Wiki!!!)
A: You should find a modern research paper or book citing this theorem. If it gives a reference for it (if not, try another reference), then go to the bibliography and check the reference. Repeat this process with this new reference: this should converge!
Edit: for this kind of theorem (which can be cited in advanced undergraduate courses which never give any reference like this), it's more difficult but in your case https://arxiv.org/pdf/1103.6166 may help. 
A: Better: they also have his collected works in five volumes,
http://oskicat.berkeley.edu/record=b14596125~S1
This link ought to show the same results it showed me, a few dozen items with author Caratheodory:
http://oskicat.berkeley.edu/search~S1?/acaratheodory/acaratheodory/1%2C5%2C39%2CB/exact&FF=acaratheodory+constantin+1873+1950&1%2C35%2C
This would probably not be the very earliest publication, but some recent papers on the arXiv refer to 

Carath ́eodory, C, Vorlesungen über reelle Funktionen, 1st e d, Berlin:
  Leipzig 1918

https://books.google.com/books?id=RV83NrXzQwEC&pg=PP1&lpg=PP1&dq=Vorlesungen+bei+reelle+Funktionen&source=bl&ots=4z4AxxWZvl&sig=fNjqEOxlvrJUtW3X886LGU-IOhM&hl=en&sa=X&ved=0ahUKEwit8aLrj9vOAhVKyWMKHcsBCIUQ6AEITDAH#v=onepage&q=Vorlesungen%20bei%20reelle%20Funktionen&f=false
which would be an entire book. There were later editions and reprints. 
Here is the link at the UCB library 
http://oskicat.berkeley.edu/record=b14989908~S1
