It's taken from the C64 music by David Dunn - you can get the SID version if you do a google search. It's Track 4 that plays in-game on the Speccy version.
The only info I could find was this (for the C64 version)...
Track 3 - Wiegenlied (Lullaby) - Johannes Brahms
But it's track 4 you want the name of, so that doesn't help.
It was also the theme music to the TV series, 'Eureka' if that helps, but Gilligan's version is a little more mixed up (Like Manic Miner's, Blue Danube)
Comments
It's taken from the C64 music by David Dunn - you can get the SID version if you do a google search. It's Track 4 that plays in-game on the Speccy version.
The only info I could find was this (for the C64 version)...
Track 3 - Wiegenlied (Lullaby) - Johannes Brahms
But it's track 4 you want the name of, so that doesn't help.
http://www.users.globalnet.co.uk/~jg27paw4/yr03/yr03_65.htm
...it's 'Trail of the Lonesome Pine' - but it doesn't sound like it to me.
Not sure about the Eureka title music either!