  We present Keck/MOSFIRE and Keck/LRIS spectroscopy of A1689-217, a lensed
(magnification $\sim7.9$), star-forming (SFR $\sim$ 16 $\rm M_\odot\ yr^{-1}$),
dwarf (log($M_\ast/M_\odot$) = $8.07-8.59$), Ly$\alpha$-emitter ($\rm
EW_0\sim138$ A) at $z$ = 2.5918. Dwarf galaxies similar to A1689-217 are common
at high redshift and likely responsible for reionization, yet few have been
studied with detailed spectroscopy. We report a 4.2$\sigma$ detection of the
electron-temperature-sensitive [O III]$\lambda$4363 emission line and use this
line to directly measure an oxygen abundance of 12+$\log$(O/H) = 8.06 $\pm$
0.12 ($\sim1/4\ \rm Z_\odot$). A1689-217 is the lowest mass galaxy at $z>$ 2
with an [O III]$\lambda$4363 detection. Using the rest-optical emission lines,
we measure A1689-217's other nebular conditions including electron temperature
($T_e$([O III]) $\sim$ 14,000 K), electron density ($n_e\sim$ 220 $\rm
cm^{-3}$) and reddening ($E(B-V)\sim$ 0.39). We study relations between
strong-line ratios and direct metallicities with A1689-217 and other galaxies
with [O III]$\lambda$4363 detections at $z\sim0-3.1$, showing that the
locally-calibrated, oxygen-based, strong-line relations are consistent from
$z\sim0-3.1$. We also show additional evidence that the $\rm O_{32}$ vs. $\rm
R_{23}$ excitation diagram can be utilized as a redshift-invariant,
direct-metallicity-based, oxygen abundance diagnostic out to $z\sim$ 3.1. From
this excitation diagram and the strong-line ratio $-$ metallicity plots, we
observe that the ionization parameter at fixed O/H is consistent with no
redshift evolution. Although A1689-217 is metal-rich for its $M_\ast$ and SFR,
we find it to be consistent within the large scatter of the low-mass end of the
Fundamental Metallicity Relation.
