The NSA has a phone number of an American citizen. No calls have been placed to or from terrorists on that number. There is no 'serious indicator' that the owner of that number has anything to do with Al Qaeda. However, the number has come up in some type of electronic surveillance program - whether it be data mining or Echelon or whatever, and it has ended up on a list. Perhaps it was called by someone who called another number that is in a terrorist rolodex. Perhaps there is some other tangential connection that would be, in and of itself, evidence of absolutely nothing untoward on the part of the citizen. The government determines it wants to tap all numbers that have been 'captured' by the program in this way. The President says it 'has to do with national security' and orders it.
Is a warrant required? If they proceed to do so without a warrant, is it a violation of law?
I am not for massive monitoring of people's numbers, its not only not "reasonable" but it is also a serious waste of resources that should be narrowly targeted.
So I would say that your scenario should lead those responsible for implementing the program to drop you off the watch list in short order. The reason you got on the list is important though. If it's tangental, as you say, then I would monitor your number to determine if you are continuing to be tangental to known terror #s or if you are only calling your aunt Edna. If you keep showing up in the loop, then you should tapped. If not, then dropped. Are there instances where the feds have monitored totally innocent people. Yes. Should they do everything they can to avoid that mistake, yes. Is it a crime for them to make that mistake. No.
A good example of that would be if we had been monitoring Atta (like we sure as hell should have been) and he had an american girlfriend. The Girlfriend is checked out and is clean as far as her known history goes. But knowing Atta was on a terror watch list, would I condone monitoring the GF's calls? Yes.
Does the occasional mistake make me change my mind and concur that warrants are needed? No
Yes and yes.