The pronunciation of Charles Xavier's family name ("egg-savier") is distinctly odd and counterintuitive. Not having read the comics as a kid, when I watched the movies, I was convinced for a long time that his name was "Charles X. Xavier" (Charles Xavier Xavier? That wouldn't be even close to the weirdest name some real world parents choose for their kids). Is there an in-universe explanation for why it this particular pronunciation exists?
This is speculation, but since you asked for an in-universe explanation, one logical explanation does exist:
The mutant association with the letter X.
The X-Men, the mutant X-gene, even "X"-avier himself as a prominent mutant advocate, etc. In-universe, mutants identify a lot with the letter X.
It would therefore make sense for a prominent mutant, either by others or even by himself, to alter the pronunciation of his name to make a political statement. Take as an example, the members of the African American community who changed their names to Muslim names during the civil rights movement, most famously the switch from Cassius Clay to Muhammad Ali. These were political statements being made using a person's name, and they're not the only example.
Again, it's definitely speculation, but it's not hard to imagine a young Charles "Zavier" deciding to pronounce his name as "Charles X-Savior" as he takes up the mantle of Mutant Rights.
Alternatively, "Charles EX-avior" might have started as a insulting deliberate mispronounciation by anti-mutant elements, (e.g. imagine a KKK member sneering at a civil rights advocate named "Danny Brickwell" by calling him "Danny Black-Well"), which was picked up and turned into a positive term by the mutant community, similar to how the term "Obamacare" went from an insult to a term used by both sides.
Either way, given Xavier's position within the mutant community, it's likely that the emphasis on the letter X in his name isn't a coincidence.
Wiktionary claims that "Zavier" is the correct pronunciation in both British and American English. It lists "Egs-avier" as an alternate pronunciation specific to the X-Men franchise.
I haven't found a source for this -- but I'm sure I've heard the "Egs-avier" pronunciation used in North America, particularly for the name "Francis Xavier". In "Francis Xavier" (referring to the Catholic saint, or people named after him), this pronunciation serves to emphasise that the name is two separate words -- otherwise it would sound more like "Francizavier".