Mount it somewhere in the file system, format it (or does one have to format first -- you'll likely want it to be in ext3 or ext4 rather than native FAT32 so Linux protection and modes are active). Depending upon what you need storage for, you might move all of /home to the SD card, or partition it for /home and /var, and mount those partitions over the original.
You'd want to make the mounting operation automatic via lines added to the fstab file. Maybe need to set privileges so user accounts can access.
debian@beaglebone:~$ cat /etc/fstab
# /etc/fstab: static file system information.
#
/dev/mmcblk1p1 / ext4 noatime,errors=remount-ro 0 1
debugfs /sys/kernel/debug debugfs defaults 0 0
debian@beaglebone:~$ ls /media/
debian@beaglebone:~$ sudo mkdir /media/sdcard
[sudo] password for debian:
debian@beaglebone:~$ ls /dev
apm_bios i2c-1 null tty tty27 tty46 tty8 vcs2
autofs i2c-2 ppp tty0 tty28 tty47 tty9 vcs3
bc_example iio:device0 ptmx tty1 tty29 tty48 ttyGS0 vcs4
block initctl ptp0 tty10 tty3 tty49 ttyO0 vcs5
btrfs-control input pts tty11 tty30 tty5 ttyO1 vcs6
bus kmem pwm tty12 tty31 tty50 ttyO2 vcsa
char kmsg random tty13 tty32 tty51 ttyO3 vcsa1
console log rfkill tty14 tty33 tty52 ttyO4 vcsa2
cpu_dma_latency loop-control rtc tty15 tty34 tty53 ttyO5 vcsa3
cuse mapper rtc0 tty16 tty35 tty54 ttyS0 vcsa4
disk mem shm tty17 tty36 tty55 ttyS1 vcsa5
dri memory_bandwidth snapshot tty18 tty37 tty56 ttyS2 vcsa6
fd mmcblk1 snd tty19 tty38 tty57 ttyS3 vhci
full mmcblk1boot0 spi tty2 tty39 tty58 ttyS4 watchdog
fuse mmcblk1boot1 spidev1.0 tty20 tty4 tty59 ttyS5 watchdog0
gpiochip0 mmcblk1p1 spidev1.1 tty21 tty40 tty6 ubi_ctrl watchdog1
gpiochip1 mmcblk1rpmb spidev2.0 tty22 tty41 tty60 uhid zero
gpiochip2 mqueue spidev2.1 tty23 tty42 tty61 uinput
gpiochip3 net stderr tty24 tty43 tty62 urandom
hwrng network_latency stdin tty25 tty44 tty63 vcs
i2c-0 network_throughput stdout tty26 tty45 tty7 vcs1
{Insert SD card -- mine has a copy of the full OS since I normally boot from it}
debian@beaglebone:~$ ls /dev
apm_bios initctl pts tty14 tty36 tty58 uhid
autofs input pwm tty15 tty37 tty59 uinput
bc_example kmem random tty16 tty38 tty6 urandom
block kmsg rfkill tty17 tty39 tty60 vcs
btrfs-control log rtc tty18 tty4 tty61 vcs1
bus loop-control rtc0 tty19 tty40 tty62 vcs2
char mapper shm tty2 tty41 tty63 vcs3
console mem snapshot tty20 tty42 tty7 vcs4
cpu_dma_latency memory_bandwidth snd tty21 tty43 tty8 vcs5
cuse mmcblk0 spi tty22 tty44 tty9 vcs6
disk mmcblk0p1 spidev1.0 tty23 tty45 ttyGS0 vcsa
dri mmcblk1 spidev1.1 tty24 tty46 ttyO0 vcsa1
fd mmcblk1boot0 spidev2.0 tty25 tty47 ttyO1 vcsa2
full mmcblk1boot1 spidev2.1 tty26 tty48 ttyO2 vcsa3
fuse mmcblk1p1 stderr tty27 tty49 ttyO3 vcsa4
gpiochip0 mmcblk1rpmb stdin tty28 tty5 ttyO4 vcsa5
gpiochip1 mqueue stdout tty29 tty50 ttyO5 vcsa6
gpiochip2 net tty tty3 tty51 ttyS0 vhci
gpiochip3 network_latency tty0 tty30 tty52 ttyS1 watchdog
hwrng network_throughput tty1 tty31 tty53 ttyS2 watchdog0
i2c-0 null tty10 tty32 tty54 ttyS3 watchdog1
i2c-1 ppp tty11 tty33 tty55 ttyS4 zero
i2c-2 ptmx tty12 tty34 tty56 ttyS5
iio:device0 ptp0 tty13 tty35 tty57 ubi_ctrl
debian@beaglebone:~$ sudo mount /dev/mmcblk0p1 /media/sdcard
debian@beaglebone:~$ ls /media/sdcard
bbb-uEnv.txt boot etc ID.txt lost+found mnt opt root sbin sys usr
bin dev home lib media nfs-uEnv.txt proc run srv tmp var
debian@beaglebone:~$ df
Filesystem 1K-blocks Used Available Use% Mounted on
udev 220096 0 220096 0% /dev
tmpfs 49496 5508 43988 12% /run
/dev/mmcblk1p1 3704040 2293176 1202992 66% /
tmpfs 247476 0 247476 0% /dev/shm
tmpfs 5120 4 5116 1% /run/lock
tmpfs 247476 0 247476 0% /sys/fs/cgroup
tmpfs 49492 0 49492 0% /run/user/1000
/dev/mmcblk0p1 7572696 2296812 4917124 32% /media/sdcard
debian@beaglebone:~$