Everyone else is just guessing. The real reason is that these were three-letter abbreviations from IBM assembly language:
CAR = Contents of the Address part of Register
CDR = Contents of the Decrement part of Register
This kind of TLA is common in assembly languages and LISP just inherited this because early implementation used these literal instructions.
No, everybody else is talking about short identifiers in general which is what the GP asked about. For car and cdr specifically you’re right, of course.
I also love how the register names don’t even have ANY mnemonic relation to what the contents mean in Lisp. Address and Decrement? Totally arbitrary. It’s like having a high-level language where function parameters are always names rsi, rdi, and so on. Because obviously "destination index" means "first argument"!