[Sony Xperia T, TL, TX, V] sjanzeir:
SOLVED!

SOLVED! Here's how it turned out:

I found this...

http://xdaforums.com/showthread.php?t=2798411

... which didn't work - at the first few attempts:

a. Installing Towelroot from internal storage using the built-in file manager seemed to be in conflict with... well, Towelroot being already installed.

b. Chainfire's superSU kept giving me an error message.

So I figured I should uninstall both using the phone's app manager and try the instructions in the link above from scratch, and voila! Here's the log:

01/000/2014 16:00:26 - INFO - Device connected with USB debugging on

01/000/2014 16:00:27 - INFO - Connected device : Sony Xperia V

01/000/2014 16:00:27 - INFO - Installed version of busybox : N/A

01/000/2014 16:00:27 - INFO - Android version : 4.3 / kernel version : /system/bin/sh: /data/local/tmp/busybox1: Text file busy / Build number : 9.2.A.2.5

01/000/2014 16:00:27 - INFO - Checking root access

01/000/2014 16:00:41 - INFO - Uninstalling com.geohot.towelroot

01/000/2014 16:00:41 - INFO - Root access denied

01/000/2014 16:00:43 - INFO - Uninstalling com.geohot.towelroot

01/000/2014 16:00:44 - INFO - Uninstalling com.geohot.towelroot

01/000/2014 16:00:44 - INFO - Uninstalling com.geohot.towelroot

01/000/2014 16:00:45 - INFO - Installing latest su binaries

01/000/2014 16:00:45 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/000/2014 16:00:55 - INFO - Installing latest su binaries

01/000/2014 16:00:55 - INFO - Installing latest su binaries

01/000/2014 16:00:55 - INFO - Root done. Cleaning files

01/000/2014 16:00:55 - INFO - Installing latest su binaries

01/000/2014 16:00:55 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/000/2014 16:00:55 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/000/2014 16:00:55 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/000/2014 16:00:55 - ERROR - error: device not found

01/000/2014 16:00:55 - ERROR - error: device not found

01/000/2014 16:00:55 - ERROR - error: device not found

01/000/2014 16:00:56 - INFO - Rebooting device

01/000/2014 16:00:56 - INFO - Device disconnected

01/001/2014 16:01:33 - INFO - Device connected with USB debugging on

01/001/2014 16:01:34 - INFO - Connected device : Sony Xperia V

01/001/2014 16:01:34 - INFO - Installed version of busybox : N/A

01/001/2014 16:01:34 - INFO - Android version : 4.3 / kernel version : 3.4.0-perf-g4c8352f-00793-g6e94e17 / Build number : 9.2.A.2.5

01/001/2014 16:01:34 - INFO - Checking root access

01/001/2014 16:01:34 - INFO - Root Access Allowed

01/001/2014 16:01:34 - INFO - Uninstalling com.geohot.towelroot

01/001/2014 16:01:34 - INFO - Stopping ric service

01/001/2014 16:01:34 - INFO - ric service stopped successfully

01/001/2014 16:01:34 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/001/2014 16:01:35 - INFO - Installing toolbox to device...

01/001/2014 16:01:35 - INFO - Pushing C:\Flashtool\custom\root\ftkit.tar to /data/local/tmp

01/001/2014 16:01:35 - INFO - Running installftkit as root thru sysrun

01/001/2014 16:01:36 - INFO - Installing latest su binaries

01/001/2014 16:01:36 - INFO - Pushing C:\Flashtool\.\devices\busybox\1.20.2\busybox to /data/local/tmp/busybox

01/001/2014 16:01:45 - INFO - Root done. Cleaning files

01/001/2014 16:01:46 - INFO - Rebooting device

01/001/2014 16:01:46 - ERROR - error: device not found

01/001/2014 16:01:46 - INFO - Device disconnected

01/002/2014 16:02:24 - INFO - Device connected with USB debugging on

01/002/2014 16:02:25 - INFO - Connected device : Sony Xperia V

01/002/2014 16:02:25 - INFO - Installed version of busybox : N/A

01/002/2014 16:02:25 - INFO - Android version : 4.3 / kernel version : 3.4.0-perf-g4c8352f-00793-g6e94e17 / Build number : 9.2.A.2.5

01/002/2014 16:02:25 - INFO - Checking root access

01/002/2014 16:02:25 - INFO - Root Access Allowed

01/002/2014 16:02:25 - INFO - Stopping ric service

01/002/2014 16:02:26 - INFO - ric service stopped successfully