Yes, at least if you are using the same XML file later and are using the same version of SoundHelix. Simply note down the logged song name and start SoundHelix with the --song-name (or -s) option with the song name (use quotes around the song name). For example, run 'run.bat -s "The angry cat" examples/SoundHelix-Piano.xml' if the logged song name was "The angry cat". The song name will be hashed to seed the random generator. You can start SoundHelix with any song name (even with song names that cannot be generated by the configured SongNameEngine) and can recreate the same song later by specifying the same song name. If you are using the SoundHelix applet, you can enter the song name directly into the song name field.
Note, however, that it is currently not possible to recreate the same sequence of songs. If you specifiy a song name, SoundHelix will regenerate and play that song, but all following songs generated will again use a random seed based on the system time.