I came up with this:
Let $S=\{(a_n)_{n\geq 1}\:|\:a_n\in \mathbb{C}, (a_n)_{n\geq 1}\text{ converges}\}$ be the set of convergent complex sequences. Then this set forms a ring under pointwise operations, with the multiplicative identity given by the constant sequence $1,1,\dots$, and the additive identity given by $0,0,\dots$.
Let $S_0=\{(a_n)_{n\geq 1}\in S\:|\:\text{for sufficiently large }n, a_n=0\}$ be the set of sequences that are "eventually $0$". Then $S_0$ is an ideal in $S$.
Define $\overline{S}=S/S_0$, and for each sequence $s\in S$, write $\overline{s}=s+S_0\in\overline{S}$. Then two sequences $s_1,s_2\in S$ are equal in $\overline{S}$ if they are "eventually identical".
Define $L:\overline{S}\rightarrow\mathbb{C}$ by $L(\overline{s})=\lim_{n\rightarrow\infty}s$. This map is well-defined, because if $\overline{s_1}=\overline{s_2}$, then $s_1$ and $s_2$ are "eventually" the same, so they have the same limit. By the familiar limit rules, $L$ is a ring homomorphism that maps unity to unity.
Let $I\subseteq\overline{S}$ be the kernel of this map. Of course, $I$ is a proper ideal.
Let $s\in S$ be a sequence that does not converge to $0$, so that means $\overline{s}\in\overline{S}\setminus I$. Then for all sufficiently large $n$, $s_n\neq 0$. So it is possible to form a sequence $s'$ such that for all sufficiently large $n$, $s'_n=1/s_n$. This means that the sequence $ss'$ is "eventually" $1$, and thus $\overline{s}$ is a unit in $\overline{S}$.
What I have just shown is that every element in $\overline{S}\setminus I$ is a unit, and since $I$ is a proper ideal, every element in $I$ is not a unit. Thus, $I$ is the unique maximal ideal in $\overline{S}$, which means that $\overline{S}$ is actually a local ring, which is nice.
Is there a name for this construction? I don't think it tells us anything new about convergent sequences of complex numbers, it's just a nice way to "repackage" what we know about convergent sequences.
The construction of this local ring is an instance of the following more general observation:
Indeed, a convergent sequence (say, with values in $\mathbb{C}$) is the same as a continuous map $$\mathbb{N} \cup \{\infty\} \to \mathbb{C},$$ where $\infty$ is mapped to the limit of the sequence. Here, $\mathbb{N} \cup \{\infty\}$ has the topology in which $\mathbb{N}$ is discrete and the sets $\mathbb{N}_{\geq n} \cup \{\infty\}$ form a basis of the neighborhoods of $\infty$. Thus, continuity is precisely the limit condition. In other words, $\mathbb{N} \cup \{\infty\}$ is the one-point compactification of the discrete space $\mathbb{N}$. A germ of a continuous function at $\infty$ is therefore a convergent sequence (defined on some $\mathbb{N}_{\geq n} \cup \{\infty\}$, but we can extend randomly on $\mathbb{N} \cup \{\infty\}$), where two such sequences are identified when they are eventually the same. The observation tells us that these form a local ring, and that the maximal ideal consists of the sequences that converge to $0$.
So, the answer to your question "Does this local ring have a name?" is: Yes, it is the ring of germs of continous functions on the one-point compactification of $\mathbb{N}$ at $\infty$.
The mentioned general observation precisely says that the stalk of the sheaf of continuous functions on a space at every point is a local ring, that is, $(X,C(-))$ is a locally ringed space.