What is the name for a vector with a single non-zero component?

I would call such vectors 'multiples of the standard basis vectors'. The standard basis vectors are normally written as $e_i=(0,0,\ldots,0,1,0,\ldots,0,0)$ where the $1$ appears in the $i$-th position, and so the vectors you are considering are equal to $ae_i=(0,0,\ldots,0,a,0,\ldots,0,0)$ for some $a$ and $i$.

Having a single non-zero component, as do, e.g., $[a, 0, 0, . . ., o]$, $[0, b, 0, . . ., 0]$, etc., is a basis-dependent property, so it does not really reflect the coordinate independence which makes many abstract vector methods so powerful. It is easy to see that, under a suitable change of basis, these vectors will no longer have exactly one non-zero component. So, I think I would adopt a somewhat prosaic description, such as vectors with a single nonzero component in the such-and-such basis; here you can think of "such-and-such basis" as a verbal variable ranging over the domain consisting of any and all bases for the vector space under consideration! ;)

The notion of standard basis vectors, as mentioned by Daniel Rust in his answer, can be a helpful one here, but in many instances it is not clear exactly what the standard basis is; for example, when we are dealing with $\Bbb F^n$ for some field $\Bbb F$, the standard is clear: $[1, 0, 0, . . . , 0]$, $[0, 1, 0, . . . , 0]$ etc. But in some contexts, there really is no standard; I am thinking here of bases for the tangent spaces on a manifold; this is a situation is which adopting a standard is not necessarily possible or advantageous.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!