"daily argument" is correct and usable in written English.
You can use it in situations when two people are having a discussion or debate that occurs on a daily basis, such as two co-workers who have opposing opinions on a specific issue. For example: Although they initially bickered over the issue, the co-workers had eventually reached a consensus after their daily argument.
Here at TechCrunch there's a daily argument in the office, on Yammer and even on the blog about the supremacy of the iPhone versus the Google-Voice-goodness of Android phones.
When I see it in the morning, I remember that you raised three boys, and I try to channel the force of your love, as my own son and I have our daily argument over pants or shorts.
In a rapid-fire monologue, Mr. Youssef ran through the subjects that touch off Egypt's daily arguments.
There were nearly daily arguments between Africa and police officers over taking down his tent, Juju said.
Doing more to help the less affluent with parenting, even if this requires "taking on the Daily Mail argument about the nanny state".
The Democratic establishment in 2005 was under assault from online activists who demanded that the party modernize its message and appeal to voters in all 50 states; now conservative blogs like The Next Right and The New Majority are making the exact same argument daily.
Whether the fiscal cliff is a real one or a manufactured one, and if real, whether it can be circumvented without lasting damage to vital welfare programs -- all that remains momentarily unresolved and as such, the legitimate subject of a daily deluge of argument.
Todd Bowles, a safety then and now an assistant with Miami, listened daily to impassioned arguments, but never considered crossing.
What I mounted in The Daily Beast was an argument.
As comedian Jon Stewart said last night on The Daily Show, Beck's argument is essentially: "Don't fix it if it's already awesome".
As Richard A. Clarke, a counterterrorism official under three presidents, wrote in The New York Daily News last week: "The argument that this sweeping search must be kept secret from the terrorists is laughable.
