Building external modules is well documented, but I found the info for rebuilding a single distro kernel module quite disjoint, I guess because it spans both distribution and kernel specific info. Now rebuilding your whole distro kernel is going to take ages due to the number of modules enabled, so the instructions below show how to quickly rebuild and install the minimum required. Note these instructions are based on rpm using distros but should be easily translatable to ones using dpkg.

The specific examples below are to patch the snd_intel8x0 driver to get sound working on Fedora 11, and enabling experimental write support on the ufs module which I used to unbrick the FreeBSD based m0n0wall router in the office.

Get kernel versions and paths for automation

kernver=$(uname -r)
kernbase=$(echo $kernver | sed 's/\([0-9]*\.[0-9]*\.[0-9]*\).*/\1/')
rpmarch=$(rpm -q --qf="%{ARCH}\n" kernel | head -n1)
rpmbase=$(rpm -q kernel | sed -n "\$s/\.$rpmarch$//p")
kernextraver=$(echo $kernver | sed "s/$kernbase\(.*\)/\1/")
arch=$(uname -m)

Install and prepare kernel source

Note the kernel-devel package only contains files required to build external modules against the kernel, not the full source.

cd /usr/src/kernels #optional
yum install yum-utils
sudo yum-builddep kernel 
yumdownloader --source kernel-$kernver 

rpm -ivh $rpmbase.src.rpm
rm $rpmbase.src.rpm #optional

rpmbuild_base=~/rpmbuild #use /usr/src/redhat for older systems

rpmbuild -bp $rpmbuild_base/SPECS/kernel.spec --target=$rpmarch
kernsrc=$rpmbuild_base/BUILD/kernel-$kernbase/linux-$kernbase.$rpmarch

Rebuild a single kernel module

config preparation

cd $kernsrc
#copy existing distro config
cp /boot/config-$kernver .config

Note if building a large module, you may want to disable debugging info by unsetting CONFIG_DEBUG_INFO in .config at this stage.

apply changes

#make config changes
sed -i 's/# CONFIG_UFS_FS_WRITE is not set/CONFIG_UFS_FS_WRITE=y/' .config

#or make code changes
$ patch -p1 < ~/alsa-fix.diff

build preparation

Note ignore the Module.symvers warning as that file should be present from the existing kernel build

sed -i "s/EXTRAVERSION =.*/EXTRAVERSION = $kernextraver/" Makefile
make oldconfig
make prepare
make modules_prepare
make SUBDIRS=scripts/mod

rebuild the module directory

Note I don't know how to build a particular module.ko within a dir, but no directory has too many modules, so the build should complete in a reasonable amount of time.

make SUBDIRS=fs/ufs/ modules

make SUBDIRS=sound/pci modules

Install and use the new module

#don't install, just use once
rmmod ufs
insmod fs/ufs/ufs.ko
mount -t ufs -o ufstype=44bsd /dev/sdb4 mnt_monowall

sudo install -m 744 snd-intel8x0.ko /lib/modules/$(uname -r)/kernel/sound/pci
killall pulseaudio
rmmod snd_intel8x0; modprobe snd_intel8x0
#relogin to start pulseaudio. Don't know how to otherwise?

Note if the module is needed in early boot it will be part of the initrd which is these days in initramfs format. To update the one in /boot manually, one can do:

mkdir tmp
cd tmp
gzip -dc /boot/initramfs-$kernver.img | cpio -i --make-directories
# copy in the new module
rm -f lib/modules/$kernver/*.bin # in case not updated by current depmod
depmod -b $PWD $(basename lib/modules/[23]*)
find . | cpio -H newc -o | gzip > /boot/initramfs-$kernver.img

Alternatively you could try the more standard

mkinitrd --force /boot/initramfs-$kernver.img $kernver

rebuild the main image

If you want to update the main kernel image rather than a module, then just build it as follows and copy it to /boot

make bzImage

optional cleanup

rm -Rf $rpmbuild_base/BUILD/kernel-$kernbase/