According to Mathworld, a Hamel basis is a basis for $\mathbb R$ considered as a vector space over $\mathbb Q$.
According to Wikipedia, the term is used in the context of infinite-dimensional vector spaces
over $\mathbb R$ or $\mathbb C$.
According to the description of the Mathematics Stack Exchange tag hamel-basis,
a Hamel basis of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.
(That is often called simply a basis, and there is no mention of infinite dimension.)
I find it difficult to reconcile these three different explanations of the term Hamel basis,
though the first two seem to be different particular examples of the third
(and for finite-dimensional vector spaces different kinds of bases are the same).
Best Answer
(1) The Mathworld definition is what Hamel had in mind, but not the way most of us use today.
(2) In modern mathematics, Hamel basis of a vector space $V$ over an arbitrary field is just a basis (i.e. a maximal linearly independent subset, or equivalently every vector can be uniquely represented as a (finite) linear combination of vectors from the set).
(3) Hamel basis is more often used in functional analysis to emphasize the algebraic nature of such basis, where other types of basis exist (e.g. Schauder basis), and because of this, sometimes it refers to vector spaces over only $\mathbb R$ or $\mathbb C$ upon which traditional functional analysis is established. For example, the orthonormal basis of an infinite dimensional Hilbert space is not a Hamel basis: It is linearly independent but not maximal. The orthonormal basis can represent every vector only if infinite linear combination is allowed (through a limit process, which is not meaningful when we are only given a vector space with no topology).