There are lots of easy ways to get a good conservative estimate of the distance. The general idea is, you put a bounding volume around your primitive and compute the shortest distance from the original point to the bounding volume. Every point on the primitive is guaranteed to be at least as far from the point as this distance.
The simplest approach is probably to use an axis-aligned bounding box. Say your triangle has vertices $\mathbf a$, $\mathbf b$, $\mathbf c$, and your point outside the triangle is $\mathbf p$. The extent of the bounding box along the $x$-axis is $[x_\min = \min(a_x,b_x,c_x), x_\max = \max(a_x,b_x,c_x)]$, and similarly for $y$ and $z$. The point $\mathbf q$ in the box $[x_\min, x_\max] \times [y_\min, y_\max] \times [z_\min, z_\max]$ closest to $\mathbf p$ has $x$-coordinate $q_x = \operatorname{clamp}(p_x, [x_\min, x_\max]) = \min(\max(p_x, x_\min), x_\max)$, and similarly for $y$ and $z$. The distance between the point and the box, then, is the distance between the points $\mathbf p$ and $\mathbf q$.
Actually, if you're going to be using this to ray-trace meshes consisting of lots of triangles, then you should really be looking into building a bounding volume hierarchy over the whole mesh as well.