It's quite simple and easy to install the Ubuntu-supported Linux GL libraries with just a small modification to your Mint installation. Why'd you want to do this? The libraries offered are generally faster, and if you watch video or play any games under X11, you know how painful an aging Intel onboard video can be.
This tutorial is assuming Linux Mint 18.1 (Sarah), and this is only useful for people with Intel On-Board Graphics. This is not for anything else.
First, edit your /etc/lsb-release to mimic it's base Ubuntu release:
; DISTRIB_ID=LinuxMint
; DISTRIB_RELEASE=18.1
; DISTRIB_CODENAME=serena
; DISTRIB_DESCRIPTION="Linux Mint 18.1 Serena"
DISTRIB_ID=Ubuntu
DISTRIB_RELEASE=16.04
DISTRIB_CODENAME=xenial
DISTRIB_DESCRIPTION="Ubuntu 16.04 LTS"
Then, download the (Linux Graphics Update Tool)[https://01.org/linuxgraphics/], at the time of this writing, it's 2.0.3. Install this with dpkg/etc. You'll note that it complains of two things, a missing library which is not available on Sarah, and a pack of fonts that are.
Go to your favorite Ubuntu mirror and get the library it's asking for- at the time of this writing, it's build libpackagekit-glib2-18, which is currently available as "libpackagekit-glib2-18_1.1.4-3ubuntu1_amd64.deb". Download, and install.
Finally,
apt-get -f install
will get the 30MB font pack installed that it wants.
Now, you can open a terminal, and run the actual installation/upgrade package:
intel-graphics-update-tool
This will download and install known good versions of libraries, and use DKMS to rebuild anything necessary for your installation. Step through to the end, and you can view the report to see what it did.
Let it reboot, and ensure that your system works- you'll be able to tell by having your framebuffer changed from VESA to an Intel framebuffer in your X logs.
Finally, change your lsb-release back (I prefer to keep, but comment out the Ubuntu data so it's available later.