First of all, your documentation mentions the path but that need to be adjusted. It is based on the current location. So, if I am standing in a current directory, I should not provide the path but only from that point. So, having said that, if the directory I wish to rename the file is the one I am currently in, I only have to provide the file name.
IAC, this still does not work. See this partial log. You can see that after the upload, the file is there. You can also see that your Rename() method is trying to rename the file exactly as it was uploaded. But, there is an error:
2012-06-21 11:22:48.622 INFO Sftp(7)[1] Command: SSH_FXP_REALPATH (1, '.')
2012-06-21 11:22:48.731 INFO Sftp(7)[1] Response: SSH_FXP_NAME (1, 1 item)
2012-06-21 11:22:48.731 INFO Sftp(7)[1] Info: Current directory is '/'.
2012-06-21 11:22:48.731 INFO Sftp(7)[1] Command: SSH_FXP_REALPATH (2, '/FirstDirectory/seconddirectory/')
2012-06-21 11:22:48.825 INFO Sftp(7)[1] Response: SSH_FXP_NAME (2, 1 item)
2012-06-21 11:22:48.825 INFO Sftp(7)[1] Command: SSH_FXP_STAT (3, '/FirstDirectory/seconddirectory')
2012-06-21 11:22:48.934 INFO Sftp(7)[1] Response: SSH_FXP_ATTRS (3)
2012-06-21 11:22:49.012 INFO Sftp(7)[4] Command: SSH_FXP_STAT (4, '/FirstDirectory/seconddirectory')
2012-06-21 11:22:49.121 INFO Sftp(7)[4] Response: SSH_FXP_ATTRS (4)
2012-06-21 11:22:49.152 INFO Sftp(7)[4] Command: SSH_FXP_STAT (5, '/FirstDirectory/seconddirectory/RebexTest.txt')
2012-06-21 11:22:49.277 INFO Sftp(7)[4] Response: SSH_FXP_STATUS (5, 2, 'File not found')
2012-06-21 11:22:49.293 INFO Sftp(7)[4] Command: SSH_FXP_OPEN (6, '/FirstDirectory/seconddirectory/RebexTest.txt', 26)
2012-06-21 11:22:49.402 INFO Sftp(7)[4] Response: SSH_FXP_HANDLE (6, 0x31)
2012-06-21 11:22:49.932 DEBUG Sftp(7)[4] Command: SSH_FXP_WRITE (7, 0x31, 0, 28672 bytes)
2012-06-21 11:22:50.494 DEBUG Sftp(7)[4] Command: SSH_FXP_WRITE (8, 0x31, 28672, 28672 bytes)
2012-06-21 11:22:50.541 DEBUG Sftp(7)[4] Command: SSH_FXP_WRITE (9, 0x31, 57344, 2991 bytes)
2012-06-21 11:22:50.541 DEBUG Sftp(7)[4] Response: SSH_FXP_STATUS (7, 0, 'OK')
2012-06-21 11:22:50.759 DEBUG Sftp(7)[4] Response: SSH_FXP_STATUS (8, 0, 'OK')
2012-06-21 11:22:50.759 DEBUG Sftp(7)[4] Response: SSH_FXP_STATUS (9, 0, 'OK')
2012-06-21 11:22:50.759 INFO Sftp(7)[4] Command: SSH_FXP_CLOSE (10, 0x31)
2012-06-21 11:22:50.837 INFO Sftp(7)[4] Response: SSH_FXP_STATUS (10, 0, 'Ok')
2012-06-21 11:22:51.071 INFO Sftp(7)[1] Command: SSH_FXP_RENAME (11, '/FirstDirectory/seconddirectory/RebexTest.txt', '/FirstDirectory/seconddirectory/Test.txt')
2012-06-21 11:22:51.180 INFO Sftp(7)[1] Response: SSH_FXP_STATUS (11, 2, 'File not found')
2012-06-21 11:22:51.196 ERROR Sftp(7)[1] Info: Rebex.Net.SftpException: No such file; File not found.
at gbMKS.PEQpz.eCVvJZ(1dRwomZ , Type )
at Rebex.Net.Sftp.Rename(String fromPath, String toPath)