In the Final Fantasy series, there is always a technician named Cid, and in the anime Soul Eater Not, one of the characters is named Sid. However, this name does not occur in the Western world.
What, then, is the significance of this name in Japanese pop culture?
If you look at the Cid page on the Final Fantasy wiki site you will find an old quote that explains this.
There really isn't any deep meaning to it, we just wanted to make a character that would appear in various forms in all the games. I guess I've always had a soft spot for that kind of character. Cid is like Yoda from the Star Wars series—very intelligent and wise.
Hironobu Sakaguchi in EGM2 August 1997.
This quote is then referenced on multiple other sites, including some localized Wikipedia pages. The same attribution is always given.
There is a problem though. That attribution is wrong. The August 1997 Electronic Gaming Monthly issue didn't feature any interview with Sakaguchi, ad to be fair I doubt that EGM2 ever had that type of content in the first place (it was more of a reviews/previews/walkthru magazine). The SEPTEMBER issue had a basic full FF7 walkthru but that's it.
Looking for images of the quote I was able to pinpoint another site that actually included a page scan. You can find it on this page of "The Lifestream"..
Once I got the actual scan it was easy to identify the real source. The page mentions GameFan and it then was simple enough to find the actual issue the page game from.
This is GameFan issue 5, July 1997.
So... here you have you answer to the Final Fantasy series Cid and the REAL source of the interview.
As for why the name comes out in other media too I guess you would have to look at each one separately.
While there is an historical figure called "El Cid", I doubt every anime Cid can be traced down to that.
Take for example Cid Kagenou in The Eminence In Shadow.
Cid's name has multiple layers of symbolic irony to it, with his surname Kageno coming from the Japanese word 'Kage' for shadow, and roughly translating to 'of shadows' in English. Cid itself derives from the Arabic honorific 'al-sid', which means 'lord', so when placed in the Japanese writing style, Kageno Cid roughly translates to 'Lord of Shadows'
