The X-Men have plenty of codenames, but Jean Grey has a history of eschewing her codename in favor of her birth name - something oddly unique for a mutant on Charles Xavier's superhero team. While the names of many of the merry mutants are mostly memorized by the masses, they continue to use codenames in battle even though many lack secret identities, unlike Spider-Man. To understand why Jean uses her name before her codename(s), one must understand the character's long history, starting from her very first appearance in The Uncanny X-Men #1.
Professor X's first class of X-Men consisted of Cyclops, Angel, Beast, and Iceman. Jean Grey was the latest addition to the group and brought a unique power to the team: telekinesis. The name Marvel Girl, given to her by Xavier, was serviceable if somewhat generic (at this point in comics history, both Marvel and DC had characters called Captain Marvel). It wouldn't be until writer Chris Claremont's run on the X-Men that Jean Grey would receive a name befitting her power: Phoenix.
As Phoenix, Jean Grey was incredibly powerful, but the Dark Phoenix Saga gave her a hunger for even more power, fueled in part by the machinations of the members of the Hellfire Club. Jean famously died at the end of the storyline, but upon her (perhaps inevitable) revival, Marvel revealed the Phoenix Force: a cosmic power that was an entity unto itself. Compounded with the fact that Rachel Summers (Jean Grey's daughter from a different timeline) also went by the name Phoenix, Jean reverted to using the name Marvel Girl...but as Jean was not a teenager anymore (none of the original X-Men were during this time), the name seemed incongruous. Susan Storm changed her name from the Invisible Girl to the Invisible Woman, but Jean never modified her original codename.
In the 90s, the X-Men animated series began airing on the Fox Kids Network, and the intro for every episode introduced the team of mutants along with their respective codenames...but Jean Grey was simply billed as "Jean Grey." Jean was an adult in the series and thus couldn't use Marvel Girl, and Phoenix was similarly unavailable because the events leading up to her adoption of the codename had yet to occur. The X-Men film in 2000 didn't give Jean a codename either (even as names like Storm and Cyclops were liberally mentioned throughout the script). In a case of adaptations influencing the original work, Jean Grey would rarely use codenames in the comics from that point forward (though Phoenix and Marvel Girl make the occasional appearances from time to time).
Neither of Jean Grey's best-known codenames were suitable in the long run (Marvel Girl only applies to a teenager and "Phoenix" was an elemental force rather than a simple name). Other superheroes such as the current Captain Marvel have struggled with multiple codenames, but in Carol Danvers' case, she acquired a name that already existed. Jean Grey, on the other hand, came to be known as Jean Grey through a combination of shifting identities and other X-Men properties influencing the character.