Choice C is correct. It's given that the equation $-9x^2 + 30x + c = 0$ has exactly one solution. A quadratic equation of the form $ax^2 + bx + c = 0$ has exactly one solution if and only if its discriminant, $b^2 - 4ac$, is equal to zero. It follows that for the given equation, $a = -9$ and $b = 30$. Substituting $-9$ for $a$ and 30 for $b$ into $b^2 - 4ac$ yields $30^2 - 4(-9)(c)$, or $900 + 36c$. Since the discriminant must equal zero, $900 + 36c = 0$. Subtracting $36c$ from both sides of this equation yields $900 = -36c$. Dividing each side of this equation by $-36$ yields $-25 = c$. Therefore, the value of $c$ is $-25$.

Choice A is incorrect. If the value of $c$ is $3$, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice B is incorrect. If the value of $c$ is $0$, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice D is incorrect. If the value of $c$ is $-53$, this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.