This is somewhat difficult to track because much of the work on linear algebra in 19th century was coached in the language of anything but matrices and vectors, differential equations, substitutions, bilinear forms, determinants, etc. That includes the works of Weierstrass and Jordan on "Jordan" canonical form. Moreover, the prefix "eigen" did not become conventional until 1960s.
In particular, Weierstrass, who published a version of the canonical form in 1868, two years before Jordan, used determinants and elementary divisors, see Hawkins, Weierstrass and the Theory of Matrices. However, Killing in his Erweiterung des Raumbegriffes (1884) spelled out Weierstrass's construction in terms of what we would call generalized eigenvectors for the adjoint action operator, see Hawkins, Emergence of the Theory of Lie Groups, 5.1:
"An important parameter in Killing's deliberations is thus the minimal
multiplicity $k$ of $w = 0$ as a characteristic root. The characteristic equation
corresponding to any $X = \sum_{i=1}^r e_iX_i$ is thus of the form
$$\Delta(w)=(-1)^r\big[w^r-\psi_1(e)w^{r-1}+\cdots\pm\psi_{r-k}(e)w^k\big]=0,$$
where $\psi_{r-k}(e)\not\equiv0$. In order to simplify the multiplication constants as
much as possible Killing chose $X = X'_1$ such that $\psi_{r-k}(e)\neq0$, i.e. such
that $k$ is the multiplicity of $w = 0$ in the characteristic equation for $X'_1$.
According to Weierstrass's theory, $X'_2, ... ,X'_k$ may then be chosen so that
$[X'_1,X'_i] = \epsilon_i X'_{i-1}$ where $\epsilon_i$ is $0$ or $1$. The $X'_i$, $i = 1, ... , k$, thus represent, in more modern terminology, a basis consisting of ordinary or generalized eigenvectors for $w = 0$, of the generalized null space $\mathfrak{g}_0(X'_1)$ of $\text{ad}\,X'_1$.
[...] Killing also showed that in the Jordan-Weierstrass form for $\text{ad}\,X_1$, at
least two Jordan blocks must exist. That is, some $Y\neq cX$ exists such
that $[X, Y] = 0$. He considered a "completely general" $X_1\in\mathfrak{g}$, by which he
apparently meant an $X_1$ such that $\text{ad}\,X_1$ has maximal rank or, equivalently,
minimal nullity... He then extended $X_1$ to a basis $X_2, ... ,X_k$ for $\mathfrak{g}$ that
put $\text{ad}\,X_1$ in its Jordan-Weierstrass form with Jordan blocks decreasing in
size."
The modern style matrix/vector exposition only emerged much later. As Miller remarks in the Earliest Uses:
"Modern expositions of spectral theory often begin with a matrix $A$ and introduce value $\lambda$ and vector $x$ together in the value/vector-equation $Ax = \lambda x$... This finite-dimensional case is used to motivate the treatment of differential equations and integral equations which involve infinite-dimensional spaces where the vector is now a function. The historical order of development was more or less the reverse. The polynomial equation generated from the differential equations of celestial mechanics came first, ca. 1780, then the equation was expressed using determinants ca. 1830, then the equation was associated with matrices ca. 1880, then integral equations were studied ca. 1900 until finally the modern order of topics starting from the value/vector-equation became established ca. 1940."
Textbooks that appeared in 1940s started associating Weierstrass's elementary divisors of matrices/linear transformations to their generalized eigenvectors/spaces, without the name. For example, Mal'cev's 1948 Russian textbook called them root vectors/spaces.
As for "eigen"values/vectors, generalized or not, Halmos wrote in Finite Dimensional Vector Spaces (1958):
"Almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature for what we call a proper value".
And only in Hilbert Space Problem Book (1967) he conceded defeat:"I have now become convinced that the war is over, and eigenvalues have won it".
