Is there a name for the great circle where latitude and longitude are equal? I have attempted a google search but only the equator and the prime meridian are defined in the sources I can find. ( It is of relevance in developing a map application which keeps track of latitude and longitude ).
The curve where latitude and longitude are equal is not a great circle.  But as joe khool writes in his excellent answer, it's called the curve of Viviani!  It's easy to see that the curve is not a great circle, because, using naïve spherical coordinates (in radians) $(\phi,\lambda)$ with $\lambda$ being longitude and $\phi$ being latitude (zero at equator), this curve passes through $(0,0)$, and also through $(\pi/2,\pi/2)$ which is the north pole ($(\pi/2,\lambda)$ is the north pole for any $\lambda$),  But it also passes through, say, $(1,1)$ which is not on the great circle the between previous two points.

In fact the curve you get looks like this:





Note. I plotted this by defining Cartesian coordinates in the obvious way:

$$
\begin{align}
 x &= R\cos\phi\cos\lambda\\
 y &= R\cos\phi\sin\lambda\\
 z &= R\sin\phi
\end{align}$$

and then plotting $(x,y,z)$ for $\phi = \lambda$ and $\lambda\in[-\pi/2,\pi/2]$.

An earlier version of this answer plotted $(x,y,z)$ for $\phi = \lambda$ and $\lambda\in[-\pi,\pi]$.  This means that $\phi$ takes values which are not in $[-\pi/2,\pi/2]$ of course.  I had assumed that these points would end up around the back of the planet: that you'd get a kind of 'S' which wraps around the planet, but in fact it ends up around the front of it again:



This surprised me!
