"Normal" & "exponential" are adjectives, already furnished with the adverbs "normally" & "exponentially". When a noun or noun phrase is pressed into service as an adjective there won't in general be a corresponding adverb to hand†. So "Weibull distributed" or "gamma distributed", though you sometimes see them, are already doing some violence to grammar (I'd avoid those). "Student's t distributed" seems even worse; I think because the possessive modifier is supposed to apply to the "t distribution", or if not just because it reminds us that "Student's t" is really a noun phrase‡. So don't say that.
"Student's t random variable" seems fine—here "t" is being used attributively, as an adjective; & why not? "Has a Student's t distribution" or "follows a Student's t distribution" are other options.
† Even when we've a proper adjective, like "Gaussian", we're still not guaranteed an adverb. "Gaussianly distributed"? No.
‡ I'm uncertain whether "Student's t distribution" is better parsed as "the t distribution, invented by Student", or "the distribution of Student's t statistic".