It was Cirdan, as it is assumed he awoke at Cuivienen Celeborn is known to predate the First Age, but is surely younger then Cirdan and Galadriel is immediately out of the question as she was born in Valinor amd Glorfindel doesn't really count, as even though reincarnated, he is not the same as before so, from the known Elves, Cirdan was the eldest
That means that either Cirdan or Celeborn would be the eldest of the Elves remaining in Middle-Earth.
I cannot say that Celeborn is surely younger than Cirdan. It is possible that both awoke at Cuivienen. Celeborn is described as a 'kinsman' of Thingol. I am not sure if that implies they are directly related but it does make it plausible that they both may have been at the Water of Awakening.
I will agree that it is likely however that Cirdan is the oldest.
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Therefore I say that we will go on, and this doom I add: the deeds that we shall do shall be the matter of song until the last days of Arda
However - if you want a more mysterious answer - How about the Elves that fled from Orome but were not caught by Melkor, some of them might still survive...
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Utúlie'n aurë! Aiya Eldalië ar Atanatári, utúlie'n aurë! Auta i lómë! Aurë entuluva!
of all Elves...well, none is oldest, but there are a few Elves such as for example Ingwe who awoke at Cuivienen and from all the Elves that awoke at Cuivienen, Ingwe has the most chances to be considered the eldest, as his name means "the first one". on the other hand, this name could also have this meaning in reference to his exalted position as High King of all the Elves. so as I said, Ingwe is the most likely answer to your question, but this is not sure.
Elwe had 2 brothers, Olwe and Elmo...and as you can't awaken and get 2 brothers immediately it is considered that their parents awoke at Cuivienen and that Elwe, Olwe and Elmo were all born there... so if they were born there, they were clearly not as old as Ingwe and anyway, Elwe (Thingol) was slain by the Dwarves in 502 FA, so he wasn't around in ME in the Third Age
Since the question is Eldest on Middle-earth, the eldest that we know of, is most likely Cirdan. It's still rather up in the air and unanswered whether he was one of the original elves who awoke and Cuivienen. However, he is one of the first Elves to appear on Middle-earth, and he'll remain in Middle-earth for a long, long, long, long time:
Some he [Osse] persuaded to remain; and those were the Falathrim, of Brithombar and Eglarest, the first mariners in Middle-earth and the first makers of ships. Cirdan the Shipwright was their lord.~Of Eldamar and the Princes of the Eldalie
Ulmo had listened to the prayers of the Noldor and their long sundering from the Teleri, so he bid the Teleri to come to Aman if they wished. Osse would miss the 'songs' of the Teleri as is said, so he asked and tried to get some to stay in Middle-earth. Cirdan was the Lord of the 'first mariners' in Middle-earth. So, I think we can say that he was one of the earliest Elves on Middle-earth. And we also know that he is the last of the Eldar to leave Middle-earth:
'But as for me, my heart is with the Sea, and I will dwell by my grey shores until the last ship sails.'~Appendix B: Tale of Years
Only the eldest of the Elves were recorded as having 'beards,' and Cirdan is one of these 'bearded' elves:
'Very tall he was, and his beard was long, and he was grey and old, save that his eyes were keen as stars...'~The Grey Havens
As TM explains Galadriel could not have been the oldest on Middle-earth...I don't know if much is known about Celeborn...but we know he wasn't bearded. Since, the question is who on Middle-earth, I would say it has to be Cirdan (unless there's some mystery Elf Tolkien never tells us about). I would think though that there were older Elves that were in Valinor...that is IF Cirdan wasn't one of the original 144 to awake at Cuivienen. Whether Cirdan was one of the original to awake, is still something that isn't quite clear, but I do strongly think he was the oldest on Middle-earth.
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I am Lórien, Lord of Dreams, my true name is 'Irmo' in Quenya.
indeed only the Elves that reach the third life circle are said to have beards...but then again one could say that other Elves didn't wish to grow a beard, even if they did perhaps have this possibility... and about Celeborn, there is one thing that also makes him younger then Cirdan. His ancestor was said to have been Elmo, brother of Elwe and Olwe, born at Cuivienen and so if Elmo appeared about in the same time as Cirdan, and Elmo was Celeborn's ancestor, it is clear that Cirdan was older
of all Elves...well, none is oldest, but there are a few Elves such as for example Ingwe who awoke at Cuivienen and from all the Elves that awoke at Cuivienen, Ingwe has the most chances to be considered the eldest, as his name means "the first one". on the other hand, this name could also have this meaning in reference to his exalted position as High King of all the Elves...
I don't think it's specifically stated that Ingwe awoke (as opposed to being born). Actually I should say it is stated, but one has to go back to the Lhammas of the 1930's for it. There Ingwe was not only the High-king but 'the oldest of all Elves, for he first awoke'.
In a later text Ingwe was said to have had a sister. Actually, if one looks at the 'counting legend' as revealing the truth with respect to the 72 spousal pairs, each of the Three Ambassadors would arguably not qualify as having awoken. That said, it might be noted too that according to Tolkien this tale was: 'Actually written (in style and simple notions) to be a surviving Elvish 'fairytale' or child's tale mingled with counting-lore'. The War of the Jewels
About the name Ingwe (more context perhaps): back in Etymologies we find the base...
ING- first, foremost. inga first (...) CF Ingwe prince of Elves' JRRT
My guess is that when Tolkien wrote this he thought Ingwe awoke first as in the Lhammas, noting the general timing involved and Christopher's commentary: 'The Etymologies then, reflect the linguistic situation in Beleriand envisaged in the Lhammas (see especially the third version, Lammasethen, p. 194)...' Christopher Tolkien
Here's what JRRT wrote much later, (probably) 1968, about the name:
'The first elements [ing-, fin-, el-, ol-] were often later explained as related to Quenya inga 'top, highest point' used adjectively as a prefix, as in ingaran 'high king', ingor 'summit of a mountain' ... [the other elements edited here] ... Of these the most probable is the relation to inga; for the Vanyar were regarded, and regarded themselves, as the leaders and principle kindred of the Eldar, as they were the eldest; and they called themselves the Ingwer -- in fact their king's proper title was Ingwe Ingweron 'chief of the chieftains'. JRRT The Shibboleth of Feanor
Earlier than this, but still post-Lord of the Rings, we have the Elvish fairy tale mingled with counting lore (written in 1959-60). This treatment puts the awakening of the Elves into a 'less historical' framework, and the names are indeed numerical -- Imin for the first Elf for example, Minyar 'Firsts' for the Vanyar and so on (which is also attested outside the counting legend proper it appears). So basically it had been an early-ish text where Tolkien had had it stated (and quite clearly) that Ingwe had first awoken. Later, while not addressing Ingwe specifically in the following, we have an Elvish Legend, a bit more 'hazy' treatment if you will, even if preserved in almost identical form by certain of the Elves...
'According to the legend, preserved in almost identical form among both the Elves of Aman and the Sindar, the Three Clans were in the beginning derived from the three Elf-fathers: Imin, Tata, and Enel (sc. One, Two, Three), and those whom each chose to join his following (...) ' The War of the Jewels
And it's on one copy of 'a form of this legend' that Tolkien wrote his:'Actually written (in style and simple notions) to be a surviving Elvish 'fairytale' or child's tale mingled with counting-lore'. [as already quoted above]. Ingwe is still Chief of the Chieftains, but so far I don't think there's been anything come to light that is both 'updated' (compared to the 1930's anyway) and reads so specifically like the older description found in the Lhammas.
Very interesting, I didn't know until know that Tolkien had had different ideas about this event as time passed... Of course, perhaps his latter thoughts should be taken more seriously into consideration, as he probably change his idea on this topic due to a good reason. However, if we are to look at the Awakening of Elves as just an Elvish fairy-tale, it does not bring us closer to the answer we are looking for. Cirdan's origin would, just as Ingwe's be only part of this mythical story, so none of them could be proven to be the first. So I guess in the end, Tolkien didn't want us to know the answer, and to make the awakening more misterious. He first gave Ingwe as the oldest of Elves, but later changed his mind and gave a more fairy-tale background to the story.
But at least we know that even with Tolkien's 'edited' version of the awakening of the Elves that Finwe, Ingwe and Elwe were leaders of there kindreds in the far East of Middle-earth. Thus they were not merely 'children' whilst they dwelt there and must of been of suitable age to become the leaders. Seeing as Elwe was slain by the Naugrim, and Finwe slain by Morgoth it leaves only Ingwe to rule over the Firstborn of the world.
Though I admit it doesn't state anywhere that he was the oldest.
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Utúlie'n aurë! Aiya Eldalië ar Atanatári, utúlie'n aurë! Auta i lómë! Aurë entuluva!
Since, the question is who on Middle-earth, I would say it has to be Cirdan (unless there's some mystery Elf Tolkien never tells us about). I would think though that there were older Elves that were in Valinor...that is IF Cirdan wasn't one of the original 144 to awake at Cuivienen. Whether Cirdan was one of the original to awake, is still something that isn't quite clear, but I do strongly think he was the oldest on Middle-earth.
