I had to install a few more dependencies from Xubuntu-16.04 Xenial, and I was not able to find files with the exact names that Brad used. I was able to find everything and piece it together. Here is a shell script that should work to install all dependencies and get going.
$ lsb_release -a
No LSB modules are available.
Distributor ID: Ubuntu
Description: Ubuntu 16.04 LTS
Release: 16.04
Codename: xenial
mkdir build-libxp
cd build-libxp/
sudo apt install dpkg-dev --install-suggests
sudo apt-get install dh-autoreconf quilt x11proto-print-dev libpng16*
sudo apt-get install libx11-dev x11proto-xext-dev libxext-dev libxau-dev xutils-dev
sudo apt-get install pkg-config
wget [mirror.umd.edu]
wget [mirror.umd.edu]
wget [mirror.umd.edu]
dpkg-source -x libxp_1.0.2-2.dsc
cd libxp-1.0.2
dpkg-buildpackage -uc -us -rfakeroot -b
cd ..
sudo dpkg -i libxp-dev_1.0.2-2_amd64.deb libxp6_1.0.2-2_amd64.deb
$ ./afni -ver
Precompiled binary linux_openmp_64: Jun 4 2016 (Version AFNI_16.1.18)
Perhaps that will help someone.
