Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number of $\mathbb{Q}^{\text{ab}}$-rational points of $y^3 - x^3 - x - 1$ is finite.
If the answer is negative, then a slight augmentation of the polynomial to a one that makes the limit finite or a bound on the rate of growth will be very much appreciated.
To answer your concrete question, first, your curve (or rather, its projective closure) is isomorphic to the elliptic curve 8649b1, $y^2 + y = x^3 - 8$. Already over $\mathbb Q$, it has rank two and therefore infinitely many rational points.
More generally, for any elliptic curve $E$ over $\mathbb Q$, you should be able to find a quadratic twist $E^{(d)}$ with positive rank; then $E({\mathbb Q}(\sqrt{d}))$ is infinite, and since ${\mathbb Q}(\sqrt{d})$ is contained in a cyclotomic field, your limit is infinite.
For curves of higher genus, the question might be more interesting.
If you have a hyperelliptic curve $C \colon y^2 = f(x)$, then you will get infinitely many points with rational $x$-coordinate and $y$-coordinate in a quadratic number field. Since the compositum of all quadratic fields is contained in ${\mathbb Q}^{\text{ab}}$, this shows that hyperelliptic curves also have infinitely many points over this field.
