This hack http://c-skills.blogspot.com/2010/07/android-trickery.html is almost universal.
I've compiled and tested a version for X10 mini.
- download and extract the archive...
You will need ADB from Android SDK and USB drivers for X10 mini (part of Sony Ericsson PC Companion).
Enable USB debugging on the phone in Settings>Applications>Development and connect the phone to PC via USB.
Add the Power Control widget to your active widgets on the phone, so you can quickly turn off and on the wi-fi function.
From the command line (on windows it's cmd.exe):
cd c:\exploid_x10mini (or another folder where you've extracted the files from exploid_x10mini.zip)
adb push Superuser.apk /sdcard/Superuser.apk
adb push su /sdcard/su
adb push exploid /sqlite_stmt_journals/exploid
adb push busybox /sqlite_stmt_journals/busybox
chmod 755 exploid
chmod 755 busybox
Immediately after the next command, you will need to disable and enable the wi-fi using the power widget on the phone:
enter password rootnow
prompt will change to # - now we have root
./busybox cp /sdcard/Superuser.apk /system/app/
./busybox cp /sdcard/su /system/bin/
./busybox cp busybox /system/bin/
chmod 4755 /system/bin/su
exit - out of rootshell
exit - out of adb shell
When you run the abd shell again and issue su command, the Superuser.apk will run on the phone and will ask whether to allow the root permissions...
EDIT: your possible questions might be already answered on the XDA forum.