I know EA DICE wanted to be spot on, on every aspect of the new Star Wars Battlefront game, but to my knowledge Luke Skywalker always used a blue lightsaber; in the previous games as well as the movies. Why is it that he has a green lightsaber in the game? Maybe because it's in beta?
Luke's blue lightsaber was lost when Vader removed his hand in Cloud City. The outfit he uses in the Battlefront beta is the outfit he wears at the beginning of Return of the Jedi, complete with green lightsaber.
As to why Dice chose to use this particular costume, I'm not entirely sure. Thematically, at the Battle of Hoth, Luke was in a standard orange Rebel Pilot uniform with his blue saber. If I had to speculate, I'd imagine that Dice thought that the Black uniform from RotJ would make it easier to recognize Luke from a distance, especially if there are or were plans for Rebel Pilot uniforms as an option for standard troops on that map.
As of January 27th, 2016; On the Hoth map, Luke is skinned with the costume he wears at the beginning of Empire Strikes Back, complete with his Blue Lightsaber. This may be because in this version of the Battle of Hoth, Luke is not piloting a Snowspeeder.
From an out-of-universe speculative perspective..
In the Battle of Hoth available during the beta you'll notice that the terrain is white, and both empire and rebel normal troops are wearing primarily white to blend in.
Vader always has the same black costume, which has stark contrast from the white background.
Since, in the game, Luke and Vader are meant to be similar in terms of balancing, it would be slightly unfair for Luke to be able to blend in better with everyone on the rebel side - if he had the correct costume.
So he has the black costume, complete with green lightsaber, to match Vader. This might just be temporary while they mess with more major balancing issues and continue adding assets to the game.
A recent update changed Luke to use his Hoth costume from The Empire Strikes Back, which includes a blue lightsaber, on any Hoth map:
He still uses his Return of the Jedi costume and green lightsaber on Tatooine and Endor maps (since that's what he wears on these planets in ROTJ) and on Sullust, which doesn't appear in any of the movies, but presumably isn't cold enough to wear Hoth gear.
Green is the color of the "Jedi Consular" class of Jedi. They are kind of the mediators and peacekeepers of the Jedi order but aren't afraid to get physical if the situation demands it.
Source. Refraining from drawing their lightsabers (many Consulars wielded green lightsabers) except as a measure of last resort, Consulars spent a great deal of time studying the mysteries of the Force.