We can use a local orthonormal basis of a parametrized curve to get a surface of the desired type.
A helix running around the $x$-axis has a parametrization like
$$
\vec{r}(t)=(ht,R\cos t, R\sin t).
$$
Its tangent vector can be gotten by differentiating
$$
\vec{t}=\frac{d\vec{r}(t)}{dt}=(h,-R\sin t,R\cos t).
$$
We note that this has constant length $\sqrt{h^2+R^2}$. With a more general curve this is not necessarily the case, and we would normalize this to unit length, and switch to using the natural parameter $s=$ the arc length. This time $ds/dt=\sqrt{h^2+R^2}$, and we can keep using $t$ as long as we remember to normalize.
We get a (local) normal $\vec{n}(t)$ vector by differentiating the (normalized) tangent
$$
\vec{n}(t)=
\frac{\frac{d\vec{t}}{dt}}{\left\Vert\frac{d\vec{t}}{dt}\right\Vert}=(0,-\cos t,-\sin t).
$$
As the name suggests this is orthogonal to the tangent vector (in the direction of change of the tangent). The third basis vector is the binormal
$$
\vec{b}(t)=\frac1{\Vert\vec{t}\Vert}\vec{t}\times\vec{n}=\frac{1}{\sqrt{R^2+h^2}}(R,h\sin t,-h\cos t).
$$
This is, of course, orthogonal to both $\vec{t}$ and $\vec{n}$.
The key is that we get the desired surface by drawing (3D-)circles with axis direction determined by the direction of the curve, i.e. the tangent. Equivalently, we draw a circle of radius $a$ in the plane spanned by $\vec{n}$ and $\vec{b}$. Hence we get the entire surface $S$ parametrized as
$$
S(t,u)=\vec{r}(t)+a\vec{n}(t)\cos u+ a\vec{b}(t)\sin u
$$
with $t$ ranging over as many loops as you wish, and $u$ ranging over the interval $[0,2\pi]$. In terms of individual coordinates this reads (barring mistakes due to lack of coffee):
$$
\begin{aligned}
x(t,u)&=ht+\frac{Ra\sin u}{\sqrt{R^2+h^2}},\\
y(t,u)&=R\cos t-a\cos t\cos u+\frac{ha\sin t\sin u}{\sqrt{R^2+h^2}},\\
z(t,u)&=R\sin t-a\sin t\cos u-\frac{ha\cos t\sin u}{\sqrt{R^2+h^2}}.
\end{aligned}
$$
Here's what Mathematica-output looks like with parameters $h=1$, $R=3$, $a=0.4$:
Here's the effect of the change to $a=1.0$. The lines on the surface correspond to constant values of $t$ and $u$. These are now more clearly defined.
