There really isn't one, and that's ok.

Before the debt ceiling, the law required the US Congress to approve any New debt issuance, every time the Treasury wanted to or needed to sell debt. As the US grew, and as deficit spending became more common, this naturally became quite burdensome for Congress, so they created the "Debt Ceiling" - selling debt up to a certain amount outstanding would not require continual congressional approval.

This, on its face, sounds reasonable, except for one problem: Congress has already implicitly authorized the deficit spending - and thus the issuance of new debt, through their budgeting. Debt is what necessarily fills the gap between taxes and spending.

Thus, every time we raise the Debt Ceiling, it's in order to bring it in line with levels of debt issuance that is already required, or in the slightly less contentious pre-Obama years, a higher level that we'd certainly eventually reach, in order to pay the bills the government has incurred. The only other country with a Ceiling, Denmark, sets their Ceiling so high, they haven't yet run into this problem.

Debt Ceiling advocates tend to argue that the Debt Ceiling is a tool used to bind Congress to a maximal level of deficit spending, or that it would be if we stopped raising it every time we reached it. Others complain about the mere fact that we continually issue any debt at all, citing the "household metaphor," that if you're spending more than you take in, you'll go broke. The difference here, with the way the government actually works, is that people clamor to purchase US debt. The Treasury Department has always paid its bills on time, and that allows our debt to be considered one of the safest investment vehicles in the world. There are essentially two ways we could get into trouble on that front:

(1) the Debt Ceiling isn't raised, and, due to precommited spending and taxation laws, we have to pick and choose which obligations get fulfilled. Most people expect our debt obligations to be paid first in this scenario, and thus, while we wouldn't default, we would take a hit in confidence, and many people (social security and Medicare recipients, government contractors, etc.) would not be paid. This is bad.

(2) the Debt Ceiling is raised continually, or abolished, and the government acts recklessly, to the point where investors start to run from our debt. This is a problem, as we like to use our cheap debt to fund our society and to pay off old obligations. Thus, if nobody buys our debt, we have to make the same hard choices as in part (1), with a lower income stream than expenditure stream.

Essentially, people who are in favor of the existence of the Debt Ceiling, and tend to oppose raising it, would believe that the risks for problem (2) outweigh those for problem (1), and those who are in favor of raising or abolishing the Debt Ceiling would argue that (1) happens with certainty if the Debt Ceiling binds, while (2) happens only in the worst case scenario if it does not.

In reality, though, politician's arguments for or against the Debt Ceiling generally come down not to economics, but to political wrangling over who will be seen as guilty for just having driven the economy off a cliff, or who is being reckless by promoting brinksmanship.