Once you have a bit of gear, you can win some games in Arena.
The bonus is, all five of your toons get the points, and in much less time (subject to queues of course) per match than a battleground.

With basically full honor gear, and no conquest gear...
My team was winning about one match in four.
And that's with everything optimized for battlegrounds, not arena.
Not a whole lot of difference, maybe... but could be using a 2-Hander and more Obliterates / Hungering Cold combo, could DeathGrip (Arena1, Arena2, etc).
With a low/initial rating, you can earn 1350 Conquest Points per week, from Arena (with a 1,650 point cap on the week); you earn 180 pts per win.

If the bonus honor battleground is IoC or AV, that's going to be the easiest method of getting Conquest Points.
You earn 100 for the first win, on a given day (either Random or the Bonus Honor BG, but not both).
Thereafter, you earn 50 for each win.
And can get a maximum of 1,650 points, which means either Arena + 300 BG points or all of your points from BG's.
A BG other than IoC/AV is generally much harder to win as a boxer who is undergeared; with full or close to full honor gear, most 15-mans should be winnable and possibly some of the 10-man's... but the smaller BG's are much harder with poor gearing.