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Grease might be gumming up a bit, proper way to fix it would be to carefully look inside, clean and regrease with the correct grease, but you'll probably be fine spraying a bit of GT85 in there and shifting up and down a bit too loosen stuff up, if nothing else it'll confirm that was actually the problem.
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Could also be a broken g spring.
It's actually relatively easy to strip one down, open it up to take a look. The second video below is assembly only, but I use both, as one shows some processes clearer than the other, and vice-versa.
Question.
Right hand Campy Record 10 speed lever no longer makes a clicking sound when changing down. It still “works” but feels spongy.