Many people argue against Xion as a character in Kingdom Hearts because of the fact that she's retconned. If no one remembers her, what's the point of her being in the series at all? If nothing would've changed because of her in Kingdom Hearts II, what keeps her from being a throwaway character?
But the fact is that she did have some subtle effects, even if no one recognizes their source. They fit in quite nicely with KHII, and help explain the motivations of certain characters.
Xion and her legacy had an impact on the following:
- Roxas Leaving the Organization
- Why Sora's waking up took so long
- Axel's behavior in KHII
- Organization XIII infiltrating the digital Twilight Town
- Naminé's behavior towards Roxas
Roxas Leaving the Organization
According to one of the producers, Xion was originally made to help explain Roxas's motivation for leaving Organization XIII. She does just that. The Organization's actions towards her and Roxas help show him just how much he's being used and left in the dark. While it's possible he may have eventually come to the same conclusions on his own, Xion speeds up the process. She causes him to question things about himself, who he is, the Organization and their purpose in it.
Roxas leaving the Oranization makes a huge impact on the plot of Kingdom Hearts II. It helps give a sense of irony to Sora's initial black-and-white judgement of the Organization and Nobodies.
Why Sora's Waking up Took so Long
Chain of Memories covers an undetermined amount of time. However, judging by how long Roxas slept during Days, we can guesstimate that it took around a month. Meanwhile, the amount of time it took to reassemble Sora's memories seems much longer than it should be. This can be explained by Xion's existence.
Xemnas and Naminé both state that Xion is absorbing Sora's memories. As she does so, Xion prevents Naminé from being able to reassemble them. There's evidence that Naminé attempted to try to stop the process, and even buy Xion time, as evidenced by DiZ's statement that "this has gone on long enough."
Xion's existence accounts for the long delay between the events of Kingdom Hearts and Kingdom Hearts II, without resorting to throwaway explanations.
Axel's Behavior in Kingdom Hearts II
There are some effects where Xion was less of a catalyst and more of an intensifier. Axel's behavior in Kingdom Hearts II is an example of this.
There's implications in Days that Axel's longstanding friendship with Saix had begun to wane. While open to interpretation, the sarcastic, snappy replies he gives to Saix prior to Castle Oblivion seem less cordial and more exasperated. This trend continues as Axel begins to befriend both Roxas and Xion.
As the game progresses, he becomes close enough to both of them that - when it becomes clear that Xion and Roxas cannot coexist - he finds himself unable to choose:
"I couldn't tell Xion she was wrong, and I can't bring myself to end her. But it's either that, or I have to erase Roxas. There's no future where the three of us can sit around and eat ice cream. I know Roxas will never understand. I don't want to acknowledge it myself."
- Axel, Secret Report (Day 353)
Axel doesn't just "not want to acknowledge it," he refuses to, as he shows in his confrontation with Xion in front of the Old Mansion. He vows to "always be there to bring [them] back," even though such a thing should be impossible. Roxas had just left the Organization the day before. Axel had just lost both of his best friends in the space of two days, friends who, as he told Sora in Kingdom Hearts II made him "feel like he had a heart."
In Kingdom Hearts re:coded, the Castle Oblivion chapter makes the point that when memories of those precious to us are lost, we carry the pain of that loss with us. Axel likely coped with that loss - the unknown, unremembered loss of Xion, as well of the loss of Roxas as his best friend - by focusing his attention in KHII on trying to get Roxas back.
Up until the end of Days, it's implied that the Organization didn't know where Sora was. This was changed when Xion went to the Old Mansion to speak to Naminé about returning to Sora. DiZ interrupts their conversation, stating that the Organization has found them and blaming Xion. At that point in time, the Organization's objective was to retrieve Xion, because Xemnas himself points out that with Xion under control, Sora may become redundant.
Later, however, after the memories of Xion herself had faded and Roxas captured, the knowledge of DiZ's base of operations (and Sora's location) became much more important. The Organization knew where they were, and were able to use that knowledge to try to work their way into the digital Twilight Town to get Roxas back.
Naminé's behavior towards Roxas
Xion's influence here was probably - as it was with Axel's - an intensifier, rather than a catalyst. When speaking to Naminé about returning to Sora, Xion has her promise to look after Roxas until he's ready to return to Sora the way Xion is. Given the context of Chain of Memories and her promise with Sora, it's clear promises are very important to Naminé.
She still probably would have cared for Roxas regardless, her being the Nobody of Kairi as Roxas was the Nobody of Sora. But the forgotten-but-not-gone memories of that promise she made probably helped to intensify her desire to speak to him and help him in Kingdom Hearts II.