You should be able to compile Chromium from source for ARM - there are some intructions on that here (probably)
To attempt running a newer version Chromium, first install Chromium from the repos to make sure all other packages and libaries (e.g.
libnss3) are installed.
Then run these command to download and extract the (currently)* latest snapshot of Chromium:
You should then be able to run
./chrome-linux/chrome and get Chromium running, unless like me you get this (or another) error:
./chrome-linux/chrome: /lib/arm-linux-gnueabihf/libc.so.6: version `GLIBC_2.15' not found (required by ./chrome-linux/chrome)
I think the focus by Debian has been pushed towards the next release of Debian (Jessie, the current version is Wheezy), so updated packages are less likely to turn up.
*To get the latest version, download the zip file inside the folder with the largest number here - the latest number is also shown here
I found the problem - wheezy, the currently stable version has libc6 version 2.13, and version 2.15 or above is needed for most Chromium ARM builds to work - you can see this in the above error, and the following when I tried installing chromium from a deb, which also resulted with a few other errors:
dpkg: dependency problems prevent configuration of chromium-browser:
chromium-browser depends on libc6 (>= 2.15); however:
Version of libc6:armhf on system is 2.13-38+rpi2+deb7u1.
chromium-browser depends on libglib2.0-0 (>= 2.35.9); however:
Version of libglib2.0-0:armhf on system is 2.33.12+really2.32.4-5.
chromium-browser depends on libudev1 (>= 183); however:
Package libudev1 is not installed.
chromium-browser depends on libnss3-1d (>= 3.12.3); however:
Package libnss3-1d is not installed.
An upgraded version is available in the next version of Debian, Jessie, but you will have to upgrade to that when Jessie is stable
As an alternative, you could just install
sudo apt-get install iceweasel
There is some info here as to why it is called Iceweasel, it was something to with Debian (on which Raspbian is based) not liking Firefox's branding...
Note: I seem to be only able to get version 24.5 on my Pi, but
Firefox Iceweasel still works.