The Cantor set is indeed an example: it’s homeomorphic to $\{0,1\}^\omega$, where $\{0,1\}$ has the discrete topology, and so is every basic open set in this product. Indeed, if $X$ is any discrete space, and $\kappa$ is any infinite cardinal, $X^\kappa$ has the property: every member of the obvious base for the product topology is clearly homeomorphic to $X^\kappa$. The Sierpiński carpet is a Cantor set.
The Cantor set is almost an example of a stronger property: it’s almost a space in which every non-empty open set is homeomorphic to the whole space. It actually has two kinds of non-empty open subset, compact ones, which are homeomorphic to the Cantor set, and non-compact ones, which are homeomorphic to the Cantor set minus a point and to the discrete union of $\omega$ copies of the Cantor set.
The irrationals are an example of a space with the stronger property: by an old result of Alexandroff and Uryson they are the unique topologically complete, separable, $0$-dimensional metric space that contains no non-empty compact open set, and all of these properties are inherited by non-empty open subsets. More generally, if $X$ is any infinite discrete space, $X^\omega$ is a metrizable space of weight $|X|$ with the stronger property.
If $\kappa$ is any infinite cardinal, $\{\kappa\setminus\alpha:\alpha<\kappa\}\cup\{\varnothing\}$ is a $T_0$ topology on $\kappa$ in which all non-empty open sets are homeomorphic.
If $\lambda\le\operatorname{cf}\kappa$ is also an infinite cardinal, $\{U\subseteq\kappa:|\kappa\setminus U|<\lambda\}\cup\{\varnothing\}$ is a $T_1$ topology on $\kappa$ with the desired property; when $\lambda=\omega$ it’s simply the cofinite topology.
