The "han" used in the Korean ethnonym is not the same "han" of that used in China (or for that matter, hanja script or the Han River, both of which which use 漢/汉 also used for the Han dynasty rather than 韓/韩 of the Zhou-period state you mentioned, which I'll refer to as the "Han state" to differentiate from the Han dynasty). In its earliest use for a sovereign state, one can look to the three confederacies of Byeonhan, Jinhan, and Mahan in the days of early Korea. This gets to be a bit tricky, since by the time of the Samhan there has already been significant Chinese influence on Korea; not so much by the Han state (which was located in central China), but rather by the Yan state (which conquered Liaoding) and the Han dynasty. The "han" used in Korea is said to be a very old native Korean word that was translated by the Chinese to the phonetically-similar (later identical due to linguistic convergence) 韓/韩, which had the benefit of differentiating the "han" of Korea from the "han" of the then-reigning Han dynasty, then was later returned to Korea as the adoption of hanja became the norm in the Korean courts around...apparently the 4th-6th centuries AD, if my understanding of this is accurate. This does seem to make a certain amount of sense, as this also appears in very old words such as grandfather (han'abi, later becoming har'abeoji), and has been speculated to mean "great" or "leader" by way of that which became the Mongolian (also "Altaic" insofar as the supergroup can be claimed to exist) "khan" (with reference also to the term "maripkan"/"maripgan" also used in the same time period by the Samhan, later replaced by the Chinese loanword "wang" - both of these meaning, in essence, "king"). On the other hand, I would hesitate to completely discard the possibility that the Korean "han" was already being influenced by Chinese contact even that early or earlier without stronger proof, as the early (pre-Chinese) Korean language is very spotty as far as the historical record is concerned.
At any rate, back to that. The "Han" used in the post-1897 name of Korea is very, very old; at a minimum almost as old as Korea, and very possibly as old as its people. For why it was chosen in 1897, Korean rulers have a habit of recycling names, which typically then received retronyms based on that - Joseon evokes older Gojoseon (originally Joseon until the later Joseon borrowed the name); Guryeo/Koryo (from which we received "Korea") evokes Goguryeo; and the Balhae and Goguryeo both anticipate the (admittedly short-lived) Later Balhae and Later Goguryeo. The choice of the name "Great Han Empire" was, as far as I can determine from my own cursory research, not intended to evoke Korean ties to China (for which it would have used the term for the dynasty, the people, or the script), but rather to underline its independence from China by hearkening back to the Samhan, which retained their independence even after the Four Commanderies were established. Though it doesn't seem to be stated and is thus speculative on my part, it likely helped that the Samhan were seen to have formed the seed of two of the Three Kingdoms (Mahan to Baekjae and Jinhan to Silla; the Gaya Confederacy which grew in the former lands of the Byeonhan were also annexed by Silla, and thus do not quite qualify), of which one (Silla) would become what was at the time seen as the first unified Korean state. By contrast, the only options earlier than the Samhan were Ancient (Go) Goryeo (when the Joseon had come to power by defeating the later Goryeo, and which itself had originally been inauspiciously crushed by the Chinese in alliance with Silla) or the Ancient (Go) Joseon, from which the dynasty had already poached the very name they were trying to replace.
But, I'm no expert, and my own texts are distressingly lacking on the Samhan period or, rather, quite superficial on pretty much anything before Later Silla; disappointing, but not surprising, considering they date back to the 60s and 70s. Please do take the above with, rather than a grain, more than a pinch of salt. ^_^
