  We examine the impact of baryon-dark matter relative velocities on
intergalactic small-scale structure and the 21 cm signal during reionization.
Streaming velocities reduced clumping in the intergalactic medium (IGM) on mass
scales of $\sim 10^4 - 10^8$ M$_{\odot}$. This effect produced a distinct
baryon acoustic oscillation (BAO) feature in the 21 cm power spectrum at wave
numbers $k\sim 0.1$ h/Mpc, near which forthcoming surveys will be most
sensitive. In contrast to the highly uncertain impact of streaming velocities
on star formation, the effect on clumping is better constrained because it is
set mainly by cosmology and straightforward gas dynamics. We quantify the
latter using coupled radiation-hydrodynamic simulations that capture the Jeans
scale of pre-reionization gas. The clumping factor of ionized gas is reduced by
5-10\% in regions with RMS streaming velocities. The suppression peaks $\approx
5$ Myr after a region is reionized, but disappears within 200 Myr due to
pressure smoothing. We model the corresponding impact on the 21 cm signal and
find that the BAO feature is most likely to appear at $\approx$ 10 \%
ionization. During this phase, the feature may appear at the 1 \% (5 \%) level
at $k \sim 0.1 (0.06)$ h/Mpc with an amplitude that varies by a factor of $<
10$ across a range of reionization histories. We also provide a model for the
signal originating from streaming velocity's impact on ionizing sources, which
can vary by 4 orders of magnitude depending on highly uncertain source
properties. We find that the clumping signal probably dominates the source one
unless Population III star formation in $10^6 - 10^8$ M$_{\odot}$ halos
contributed significantly to the first 10\% of reionization.
