It's just the same as using other friendly name-versions such as Jimmer (for Jim/James), Shazza (for Sharon) or Gazza (for Gascoyne). I don't think any of these is specifically Liverpudlian...in north-east Scotland, for example, Gordon is often called Gogsy, which shows a virtually identical process to the formation of Degsy..
Could it be related to the Southern (London? Cockney?) shortening names to rhyme with bell - Tel (Terence), Del (Derek). I'm sure there's another I cannot call to mind. Then there's Baz (Barry), Daz (Darren), and Shaz (Sharon).