What is the name for a vector field that is both divergence-free and curl-free?

In geometric calculus literature (see, for example, Doran and Lasenby), such a function is called monogenic. Monogenic functions are generalizations of complex analytic (or holomorphic) functions. This condition is strictly stronger than being harmonic--all monogenic functions are harmonic, but not all harmonic functions are monogenic.

The term monogenic is not restricted to vector fields, as well; a scalar field with zero gradient would also be referred to as monogenic.

You can also consult this page by Gull, Lasenby, and Doran.

Edit: Phrased in the language of geometric calculus, we define the vector derivative of a vector field $u$ as $\nabla u$, given by

$$\nabla u = \nabla \cdot u + \nabla \wedge u$$

When $\nabla u = 0$, then $\nabla^2 u = \nabla \wedge (\nabla \cdot u) + \nabla \cdot (\nabla \wedge u) = 0$ as well, fulfilling the harmonic condition.

Put musical isomorphism aside, I believe what Rahul Narain refers to is just harmonic $1$-form.

In Hodge decomposition for $k$-forms $\omega$: $$ \omega =\mathrm{d}\alpha +\delta \beta + \gamma $$ where $\gamma$ is harmonic in that $(\mathrm{d}\delta + \delta\mathrm{d})\omega = 0$, and $\delta = (-1)^{nk+n+1} \star^{n-k+1}\mathrm{d}^{n-k}\star^k$.

In the 3-dimensional case. We have the cochain complex: $$ \Lambda^0\ \stackrel{\mathrm{d}^0}{\longrightarrow}\ \Lambda^1 \ \stackrel{\mathrm{d}^1}{\longrightarrow}\ \Lambda^2\ \stackrel{\mathrm{d}^2}{\longrightarrow}\ \Lambda^3. $$ Define $\delta = \mathrm{d}^*_{k}: \Lambda^*_k\to\Lambda^*_{k-1}$ as the adjoint of $\mathrm{d}^{k-1}: \Lambda^{k-1}\to\Lambda^{k}$ with respect to the inner product. We can have somewhat a dual complex: $$ \Lambda^*_3\ \stackrel{\mathrm{d}^*_3}{\longrightarrow}\ \Lambda^*_2 \ \stackrel{\mathrm{d}^*_2}{\longrightarrow}\ \Lambda^*_1\ \stackrel{\mathrm{d}^*_1}{\longrightarrow}\ \Lambda^*_0. $$ For a harmonic 1-form $\gamma$: $$ (\mathrm{d}\delta + \delta\mathrm{d}) \gamma = (\mathrm{d}^0\mathrm{d}^*_1 + \mathrm{d}^*_2 \mathrm{d}^1)\gamma = 0. \tag{1} $$ Note: $$\mathrm{d}^1 = \nabla \times, \quad\mathrm{d}^*_1 = (-1)\star^{0}\mathrm{d}^{2}\star^1 = -\nabla \cdot$$ (1) is: $$ \nabla \times (\nabla \times \gamma) - \nabla (\nabla \cdot \gamma) = 0.\tag{2} $$ We can say a curl-free and a divergence-free vector field is harmonic under musical isomorphism for $$ \nabla \times \gamma = 0\;\text{ and } \nabla \cdot \gamma = 0\Longrightarrow\nabla \times (\nabla \times \gamma) - \nabla (\nabla \cdot \gamma) = 0 . $$

I am guessing the wikipedia page was using "Laplacian vector fields" in that (2)'s left side is actually the vector Laplace operator (or Laplace-Beltrami) acting on a vector field.

For references, we use this term a lot in computational geometry, a field which inherits a lot of terminologies from vector calculus, it is like almost a tradition that saying a vector field is harmonic means it is curl-free and divergence-free w/o citing anyone's book.