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It's not a recognised English expression -- it's a colloquial, humorous one which the author may even have invented, based on similar colloquial, humorous expressions such as: He's got....
* a case of the shakes (=he's shaking)
* a case of the Mondays (=he's feeling miserable because it's his first day back at work after the weekend)
* a case of the sniffles (=he's got a cold)
* a case of the stomach monkeys (=he's feeling apprehensive about some forthcoming event -- eg, taking an exam). They're designed to (jokingly) make it sound as though the person has a professionally-diagnosed medical illness, as in the following:
* a case of influenza
* a case of pneumonia
* a case of heart disease
which show the correct use of 'a case of ......'. So 'he's got a case of the weirds' will just mean 'he's experiencing weird feelings/sensations' -- as you do when you have a hangover.
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