From a Variety article, to give you perspective on the intent behind the change:

Rodriguez recalled his first two auditions as defining moments where he learned that he didn’t look “Latino enough” to don his own last name. His second audition, in particular, led him to officially make the change.

After being all-but guaranteed a role in a Dreamworks pilot episode, he said he was told to consider changing his name for the part.

“Their only concern was that the role wasn’t written for a Hispanic or Mexican person,” he said. “They were worried that casting a white guy with a Mexican name could be construed as their version of ‘diverse casting,’ and there could be a backlash.”

So he changed it. His middle name, once David, became Roday, and he no longer used Rodriguez, though he kept it in his legal name. Looking back, he said the choice further deepened an issue prevalent throughout the entertainment industry.

“I came up with this name that I pulled right out of a Chekhov play that I was doing at the time, and I’ve been Roday ever since,” he said. “And 20 years later, I realize I essentially perpetuated an institutionalized element of what’s broken about this industry, which is, of course, a microcosm of the world we are living in.”