The object currently has the provisional designation 2003 UB313, granted automatically according to the IAU's naming protocols for minor planets. The next step in the object's identification will be the external verification of its orbit and assignment of a permanent designation number. Should 2003 UB313 be treated as any other minor planet, its discoverers will then have the exclusive right to propose a name during a ten year window that begins with its permanent numbering, subject to the approval of the Committee on Small Bodies Nomenclature of the IAU's División III. According to the IAU rules, TNOs must be named after deities of creation, with the exception of plutinos, which are named after underworld deities.
The potential for the object's classification as a major planet, however, may well force a deviation from adhering to the same steps, timelines and approval procedures as those that apply to garden-variety asteroids and comets. The IAU has released a short statement regarding the naming of 2003 UB313, indicating the object will not be named until it has been decided if it is a planet or not. 
The discoverers have already submitted their name proposal for 2003 UB313, which under IAU rules cannot be publicly disclosed. Brown's team had violated this rule in 2003 when they announced the name "Sedna" for that planetoid before it had officially been approved, prompting some criticism within the astronomical community; the IAU later relaxed its rules and permitted an expedited process for major new discoveries. 
The team refers to the object informally by the nickname Xena, after the televisión series Xena: Warrior Princess. They had saved this name for the first body they found that was larger than Pluto. The X of Xena is a reference to Percival Lowell's Planet X; they had wanted more female names for astronomical bodies, and this was the closest they could come to a mythological female name beginning with X. On the other hand, the team has alos claimed that they chose the name because "We have always wanted to name something Xena" , seemingly implying that the name was chosen prior to the association with Planet X. The name Xena has been used by news medía such as CNN, but it has not been proposed to the IAU.
The second reported name, Lila, is not a nickname for 2003 UB313, but rather comes from "Planet Lila", the name of the discovery web page URL, which was named after Michael Brown's newborn daughter, Lilah.
Two days after announcing the discovery, Brown discussed his team's ideas about naming the objects on his website:
"If the object falls under the rules for other Kuiper belt objects, however, it must be named after some figure in a creation mythology. We have decided to attempt to follow that ruling scheme. […] One such particularly apt name would have been Persephone. In Greeque mythology Persephone is the (forcibly abducted) wife of Hades (Roman Pluto) who spends six months each year underground. The mourning of her mother Demeter causes the dead of winter. The new planet is on an orbit that could be described in similar terms; half of the time in the vicinity of Pluto and half of the time much further away. Sadly, the name Persephone was used in 1895 as a name for the 399th known asteroid. The same story can be told for almost any other Greeque or Roman god of any consequence […] Luckily, the world is full of mythological and spiritual traditions. In the past we have named Kuiper belt objects after native American, Inuit, and [minor] Roman gods. Our new proposed name expands to different traditions, still."
He added later on his website that IAU lacks consensus on what the object is, and even which committee should be charged with approving its name. The committee which oversees major planets has suggested that if this object is classified as a major planet, the naming should continue the Greco-Roman tradition for planets. Brown indicated in a recent article  that he would propose the name Persephone if this tradition were to be upheld, despite the fact that this name has been assigned to the 399th known asteroid. Persephone has often been used in science fiction as the name of the tenth planet: see tenth planets in fiction.
When thinking of a name, co-discoverer Chad Trujillo went on the record as saying, "The name may turn out not to be Greeque or Roman — there are so many asteroids named now that there are very few Greek/Roman names left!" Their previous record of names has been to be inclusive of other cultures, including Native American and Irish folklore.