How to divide aleph numbers
Much like I wrote in Cardinal number subtraction, if $\kappa$ and $\lambda$ are two $\aleph$-numbers, it might be possible to define division, but this definition would have to be limited and awkward.
If $\kappa$ and $\lambda$ are both regular cardinals and $\kappa<\lambda$ then every partition of $\lambda$ into $\kappa$ many parts would have to have at least one part would be of size $\lambda$. In a sense this means that $\frac\lambda\kappa=\lambda$. This is indeed the case with $\aleph_1/\aleph_0$, both are regular cardinals are $\aleph_0<\aleph_1$.
If, however, $\kappa=\lambda$ this is no longer defined, since $\kappa=2\cdot\kappa=\aleph_0\cdot\kappa=\ldots=\kappa\cdot\kappa=\ldots$, so there can be many partitions of $\lambda$ into $\kappa$ many parts, and in each the parts would vary in size (singletons; pairs; countably infinite sets; etc.)
When $\lambda$ is a singular limit cardinal, e.g. $\aleph_\omega$ this breaks down completely, since singular cardinals can be partitioned into a "few" "small" parts. In the $\aleph_\omega$ case these would be parts of size $\aleph_n$ for every $n$, which make a countable partition in which all parts are smaller than $\aleph_\omega$.
The only reasonable way I can think that cardinal division can be defined would have to consider the Surreal numbers, and the embedding of the ordinals in them. However this will not be compatible with cardinal arithmetic at all (the surreal numbers form a field).
I should also remark that your reasoning for $\aleph_1/\aleph_0$ being $\aleph_1$ is invalid. First note that neither is a real number, and that it is possible that $\aleph_1$ is much smaller than the cardinality of the real numbers (so between two natural numbers there are a lot more real numbers). Secondly, note that between two rational numbers there are also infinitely many rational numbers - does that mean $\aleph_0/\aleph_0=\aleph_0$?
However your rationale is not that far off, as I remarked in the top part of the post, if you take a set of size $\aleph_1$ and partition it into $\aleph_0$ many parts you are guaranteed that at least one of the parts would have size $\aleph_1$.
Further reading:
- Cofinality of cardinals
- Cofinality and its Consequences
- How to understand the regular cardinal?
- How far do known ordinal notations span? (Cantor normal form)
- Surreal and ordinal numbers
If you're looking for something compatible with cardinal multiplication, you'll have to deal with the fundamental problem that $\kappa\cdot\lambda=\max\{\kappa,\lambda\}$ whenever $\kappa,\lambda$ are well-orderable cardinals and at least one of them is an aleph. That sort of absorption means--for example--that $\aleph_0\cdot\aleph_2=\aleph_1\cdot\aleph_2=\aleph_2\cdot\aleph_2=\aleph_2$, so even trying to define $\aleph_2/\aleph_2$ in some way compatible with cardinal multiplication is problematic. Now, one could choose a convention for $\lambda/\kappa$ in instances that $\kappa\leq\lambda$--say, for example, that it's always just $\lambda$--but trying to compatibly define it when $\kappa>\lambda$ is fruitless.