Saying 'This argument is frivolous' is shorthand for 'No reasonable person, having done basic research of the issue, would believe that this argument has any chance of success in court.'
As a result, 'frivolousness' is basically a combination of two factors: the meritlessness of the argument, and the number of times that the argument has been presented and lost before.
To give an example of one extreme: Suppose I argue that I should be exempt from income tax because all US currency and income tax forms have embedded nanoelectronics that monitor my thoughts and transmit hostile messages into my brain, and that thus the government has unclean hands. There's a very good chance that this particular argument has never been presented in court before. But even so, it could safely be described as a frivolous argument, because it's so unbelievably meritless that no reasonable person would believe that it has a chance of success.
To give an example of the other extreme: Suppose I argue that the Sixteenth Amendment was not properly ratified. This is an argument that, when originally made, was not frivolous, because there were indeed some irregularities in the process surrounding the Sixteenth Amendment (and others), and because the issue of what makes an amendment 'properly ratified' had not been completely worked out to begin with. But as of now, the argument has been made in court again and again and again and again and again, and it has lost at all levels (including the Supreme Court) again and again and again and again and again, and there is no indication that the Supreme Court has any intention of changing its mind on the subject. So it's a frivolous argument, because no reasonable person having looked at the cases would believe that it has a chance of success.
Most of the frivolous arguments listed in documents like "The Truth About Frivolous Tax Arguments" are somewhere in the middle. They're frivolous because the arguments themselves have little or no intrinsic merit, and because the courts have considered and rejected the arguments.