Is there a name for the result of summing the bins of an FFT?
It has the name "first element of the time domain vector times the length of the vector".

reason:

With the DFT of a $N$ long sequence $y[n]$ being

$$Y[k] = \sum_{n=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N}\text,$$

your sum of FFT bins is

\begin{align}
\sum_{k=0}^{N -1}Y[k] &amp;= \sum_{k=0}^{N -1}\sum_{n=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N}&amp;\text{finite sums can be exchanged in order}\\
&amp;=\sum_{n=0}^{N -1}\sum_{k=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N}
&amp;\text{$y[n]$ is constant over all $k$}\\
&amp;=\sum_{n=0}^{N -1}y[n]\underbrace{\sum_{k=0}^{N -1}e^{-j2\pi k\frac{n}N}}_{=\begin{cases}N\cdot1 &amp;n=0\\0&amp;\text{else}\end{cases}}&amp;
\text{sum-orthonogonality of the DFT}\\
&amp;=y[0]N
\end{align}