The oldest elves in Arda are those who awoke at Cuivienen. In what is little more than a children's tale, the original six are named as Imin & Iminyë, Tata & Tatië, Enel & Enelyë. Then there is a gap in our knowledge, and we encounter the oldest living elves: Ingwë of the Vanyar, Finwë (and Miriel) of the Noldor, and the brothers Elwë, Olwë and Elmo of the Teleri/Lindar. To this we can add the unnamed sister of Ingwë, whose daughter Indis married Finwë. Oh, and Nowë, later called Cirdan ('Shipwright'), who is listed as kin to Elwë and Olwë. There are also Morwë and Nurwë, leaders of the Avari, and Lenwë, leader of the Nandor.
Could these elves have woken at Cuivienen? Certainly there is no impediment to Imin and Ingwë coexisting, or Tata and Finwë, Enel and Elwë - we know that the three kings of the Great March assumed that role because they had been the messengers to Valinor. Assuming the original three leaders were still alive, they would have yielded authority on the March to those who knew about the destination - and when they finally reached it, there would have been no incentive to resume the leadership.
It is notable that there are several brothers, but no parents, listed among the -wës. According to the tale of Imin et al, the Quendi awoke in pairs (husband and wife), and beyond that in groups: first six (Imin, Tata, Enel and their wives), then twelve (who joined the Vanyar), 18 (to the Noldor), 24 (to the Teleri), 36 (the Noldor again), and finally 48 (the Teleri again). Could it be that 'brothers' and 'sisters' are those who awoke in the same group (obviously excluding one's own spouse)? Breaking down the known and inferred names (and excluding the Original Six):
-Ingwë, his wife, his sister (Indis' mother), and her husband, account for four of the twelve Awakened Vanyar.
-Finwë, Miriel, Morwë (leader of the Avarin Noldor) and his wife account for four of the 54 Awakened Noldor.
-Elwë, his wife, Olwë, his wife Eärwen, Elmo, his wife, Nowë, his wife, Nurwë, his wife, Lenwë, and his wife, make together 12 of the 72 Awakened Teleri. At least eight of these - Elwë, Olwë, Elmo, Nowë, and their respective wives - must have awakened in the same group.
Of course, a major problem arises: Elwë is known to have taken a different wife, the Maia Melian. Except... while the Eldar were created in pairs, and the Original Six married in the couples they awakened in - there is no reason to assume every one of the 72 couples was a perfect match. The apparently 'destined' spouses could simply not have gotten on with each other, and so, rather than marrying (after a presumed long engagement period anyway - in Valinor the engagement lasted one Valian Year, or roughly ten years of the sun; this may well be roughly equal in length to the period of 'engagement' before the awakened Quendi discovered sex!), would have sought other spouses - or remained alone. This of course makes it possible that some of the unnamed wives (and one unnamed husband, that of Ingwë's sister) could have been of later generations - but my suspicion is that the original Quendi would find that rather strange, marrying a born, rather than awakened, elf.
And there remains one mystery: Eöl of Nan Elmoth, kinsman to Thingol, master-smith, kidnapper and abusive father. In his relations with Elu (formerly Elwë) Thingol, he doesn't come off as a young nephew or other such relation - he acts as an equal (one who has been rather cheated out of his rightful authority). So is he Thingol's brother - an awakened elf who never took a bride until Aredhel?
'Eöl' is not a name in Quenya or Sindarin. It is often theorised to be Avarin - but there is no reason for Eöl to have spent any time with the Avari. In fact, he seems to spend more time with the Dwarves of Nogrod and Belegost - dwarves who are known, on occasion, to bestow names on elves they approve of, such as Felakgundu, 'Hewer of Caves' - or Finrod Felagund, as we know him.
So could Eöl be an aftername given by the dwarves? It doesn't sound all that much like Khuzdul - but nor does it sound like anything else. It would make a great deal of sense, for an elf who seems to regard his friendship with the dwarves to be more important than his son or wife. But in that case - who was he, before he took a Dwarvish name?
Elmo, father of Celeborn, is known from a dubious source as a brother of Elwë - the unimportant one, compared to his two (or possibly three, if Cirdan is counted) famous brothers. Might he have become irritated, even angry with his lack of status, and turned away from the ungrateful Sindar, instead spending his time with the dwarves? Might he have taken up residence in a dark forest - in fact, the one where his brother met Melian - and learnt arcane arts of sword-making? Might he, in fact, have become Eöl of Nan Elmoth?
Of course, this would seem to require Elmo/Eöl to take two wives - Celeborn's mother, and Aredhel. Could a lord of the Sindar fall so far as to willingly engage in bigamy? It could have been a custom he absorbed from the dwarves, but it seems to run against much that we know of the Quendi (sort of like taking a wife by as-near-to-force-as-possible, ahem). More palatably, Celeborn could be adopted, his parents victims of Melkor's early attacks - or of the hazards of the March. If Elmo (still, at that time) took young Celeborn under his wing for the long March, he could well have called him 'son'.
It's certainly possible. And if it's true - what do you imagine he would have thought of his Celeborn marrying Galadriel of the Noldor he so despised? And does it support this theory to note that, in the Second Age, Celeborn and Galadriel founded a new nation - a nation of smiths...