I think that to say "to qualify" = "to weaken" is too simplistic. To qualify is to add limitations or conditions.
Consider the following claim: I would eat anything.
This is a blanket statement with no real qualification (limitation). After making this statement, I might think "wow maybe I went too far. Let me qualify that". Then I would make the following claim:
If I am really hungry, I would eat anything.
This is the original claim qualified (limited) by a specific condition. If I think the claim needs further limitations still, I can qualify it further:
If I am really hungry, I would eat anything that is properly cooked.
To qualify is to add limitations or conditions. In RC, if a claim is presented, the author of the passage (or another expert within the passage) can qualify the claim by showing that something else should be taken into consideration, or that the claim has some limitation. This is likely simplified as "to qualify" = "to weaken".
For example, an expert may argue that "Tigers are solitary animals". The author of the passage (or an answer choice to a question) might specify that "Tigers must meet to breed" or "Females care for their cubs for 2 years". These don't destroy the initial claim, but they qualify it (add limitations to its accuracy). Again, this is likely simplified as "to qualify" = "to weaken".