It is possible Solon concocted his story, but unlikely. There is however good reason to believe he embellished it significantly. Perhaps he didn't embellish so much as change the emphasis. He was a statesman and lawmaker who fought the moral decline in Athenian society. Although unsuccessful during his time in office, Solon is considered by many to have laid the foundations of democracy. There is understandably no documented record of Atlantis before Solon - ignoring possible ambiguous references from Homer and Hesiod - not in Greek literature anyway.
Around 594 BC Solon was chosen as archon (chief magistrate) in Athens. Solon implemented reforms largely unpopular with the ruling class. He travelled abroad for 10 years so influential Athenians couldn't induce him to repeal any of his laws. First off he went to Egypt and it was there the story of Atlantis came to light. Solon visited the Pharaoh, Amasis II (reign 570-526 BC), and spent some time with two Egyptian priests discussing philosophy. The priest Sonchis of Sais is attributed as the source of the history of Atlantis.
Note: Sais is stated as the birthplace of the Pharaoh Amassis II which adds a little weight to Solon being there, although Sonchis being the priest who related the story is contested by some.
The Greek Dark Ages (c.1200-800 BC) blotted out ancient history, as surmised by the Egyptian priest in the Timaeus Dialogue (see Literature [1c]). Plato's account doesn't name the priests. Sonchis speaks of a great victory by Athens in defeating the all-conquering Atlanteans. Although the Dialogue is consistent, there are question marks against certain aspects. It is conceivable Solon twisted the story to put Athens in a better light, so he could best relate the moralistic tale without further antagonising those in power. On the other hand Sonchis might well have done that himself, but if he had been discussing philosophy with Solon it is unlikely.
There is no record of Timaeus anywhere else. He is Italian according to the Dialogue and his story of the creation has similarities with the biblical version in Genesis. Critias is likely genuine, as is Hermocrates the third speaker who never spoke, or whose dialogue is lost. Both are referenced by others from the time. The existence of Socrates is not in doubt. What is in doubt is if these dialogues were a coming together. It is almost certain the part of Socrates in the dialogues are assumptions by Plato, on how he perceived his mentor would respond. This could also be a cover story for his own protection. Nevertheless, although most date the Dialogues to 360 BC, some 39 years after the death of Socrates, it doesn't mean such a meeting never took place. My personal view is that Plato met with his fellow protagonists but at different times and simply put together a series of dialogues he had with them individually.
The more we look at it, the more we see evidence of some kind of fabrication. Was Plato responsible for carrying out a deception or merely recording the thoughts of others? Nor did he always get it right as we see with his Pythagorean notes in Republic, which are nonsensical in their ambiguity. In Critias the Gods are barely mentioned apart from Poseidon, Though purported descendants of the Gods, the mortality of the Atlanteans is clear. At the end of the incomplete Dialogue there is mention of Zeus as 'God of Gods' before the text stops abruptly. Did Plato see he was painting himself into a corner?
In the Dialogues there is mention of an absentee who was taken ill, and then there is Hermocrates who never managed to speak, whose dialogue was lost, or was simply not written. Hermocrates could be key. He was a general who repelled Athens in Sicily and advised Sparta. His part can be surmised but the real issue is the time of his death, 407 BC and the fact he had returned to Sicily in 408 BC. There is divided opinion on Plato's date of birth. Taking the earliest estimates he would be 19/20 years old when Hermocrates returned to Sicily, taking the latest he would have been 16/17.
Socrates was a member of the Boule (council of citizens) who debated the fate of the generals after the Battle of Arginusae (406 BC). The means the meeting would still be plausible, Critias was one of the Thirty Tyrants installed to govern Athens after the fall in 404 BC, Timaeus has always been a mystery. Perhaps the meeting took place and during the writing Plato heard of the death of Hermocrates. Academics will argue the Dialogue was among Plato's later works but many an author has old drafts they only publish much later. When all said and done, I am perhaps a little harsh on Plato, he is after all just an author.
There is of course much more to Plato than the scant mention here, and I could dissect all his works, but all I'm trying to establish is how much credence we can give him with regard to the Timaeus and Critias Dialogue. It is important to look at things objectively and inconsistencies must be addressed. Having said that, even the implausible cannot be dismissed out of hand, often fact is stranger than fiction.
Irrespective of the conspiracy theory and the ideology behind it influenced by Solon / Socrates / Plato, the main body of evidence suggests the basis of the Atlantis story is historical fact.