I know this is an old post, but it still comes up in searches, so I'm going to give a solution.
Most distros supply memtest86+. It is not compatible with UEFI, because UEFI does not support 16 bit . However, PassMark Memtest86 (ver. 5 or later) supports UEFI.
First, obtain the memtest files. Download the USB boot image. http://memtest86.com/download.htm Next, extract the files from the image. I burned the image to a flash drive to facilitate installing on multiple computers.
Second, install memtest86. I made a 500MB partition at the end of my HDD, and formatted to FAT32. Then copied all files from flash drive to this partition. Note: it is possible to install to your EFI directory, but files need to be renamed to avoid overwriting existing boot files.
Third, configure grub2-efi menu entry. There are many ways to add an entry, but the safest way that survives update-grub follows. Using root privileges create file /etc/grub.d/39_memtest, and set it executable. (Note: changing the number in the file name changes the position it appears in the grub menu. DO NOT make it lower than 6 to insure that header and theme run first)
Next make the contents of the file:
#!/bin/sh
echo "Adding Memtest86" >&2
exec tail -n +3 $0
menuentry "Memtest86" {
chainloader (hdx,y)/efi/boot/bootx64.efi
}
EOF
*Replace (hdx,y) with your appropriate HDD and partition information.
Then run update-grub You should now have a menu entry that boots into Memtest86