Provided your bootloader is GRUB2 and is installed on sda1, you probably have a file  .../sda1/boot/grub/grub.cfg (or similar location).
Crate a file next to  grub.cfg  with the file name  custom.cfg .
GRUB2 will by default search for this file and use its content. This way you can add additional boot options that won't get removed when your Lubuntu install rewrites grub.cfg after an update.
If you want to boot Tiny Core on sda2, using the structure I suggested above, add this to custom.cfg:
menuentry "Tiny Core 6.x" {
	insmod ext2
	linux (hd0,msdos2)/boot/tce/boot/vmlinuz  quiet loglevel=3 tce=sda2 desktop=openbox
	initrd (hd0,msdos2)/boot/tce/boot/core.gz
}