They're easy to produce, have an attractive physical appearance, provide an immediate motivation for players, and their method of delivery helps foster a period atmosphere. While useful, for most purposes they're still just a prop- a physical object that just happens to contain information to nudge players into a pre-planned story. The message is what matters, not it's method of delivery. In the vast majority of cases the details of how the telegram was sent have little impact on the story.
But, sometimes, the little details behind a telgram do matter. For those occasions, the wonderful "Telegraph Office" website has the text of a 1928 pamphlet entitled "How to Write Telegrams Properly" . It's a fascinating look at the ins and outs of the system, from the basics of conveying messages in as few words as possible to the things an adventurous Mythos investigator or pulp protagonist needs to know. How can you make sure messages sent to your home office arrive at your field location? How do you send money? How do you confirm receipt of a message? How do you wire money?
"The procedure is simple. A person wishing to send a sum of money by wire merely calls at the telegraph office, fills out an application blank, and pays the clerk the amount to be sent and the fee for its transmittal. The telegraph companies have a secret code which they use in directing their agent in the distant city to make payment to the person designated. The payee is notified to call at the office for a sum of money, or a check is sent to the payee, as may be directed. It is optional with the sender of the money order, whether the payee shall be required to identify himself absolutely or whether identification shall be waived. The Western Union Telegraph Company alone handles more than $250,000,000 annually in telegraphic money orders."
Read it and you'll know exactly what to do the next time you're shivering in the middle of nowhere covered in slime and gore. Well, maybe not everything you need to do, but at least you'll be able to tell Armitage you survived and need some money for new clothes and a hotel room.