Two similar mythological roots suggested are that the town is named after either the mythical King Leir (as Leicester is said to be), who seems to have been a genuine King somewhere in Britain, possibly Northumbia, though no one is quite sure; or after the Welsh mythical figure Lyr, who’s children were turned into Swans, ( something that has been posited as an original of the Liver Bird). A problem with these and others is that the settlement was little more than a farming or fishing outpost of West Derby until King John created the town in the 1200s, and the name pre-dated that, so it is highly unlikely the name relates to a settlement at all.
Two more are that it was ‘Lavapool’, after the seaweed lava which was very widespread as a foodstuff, or a corruption of ‘Elverpool’, after the large number of ells (elvers) the Mersey was known for in medieval times.
An interesting theory which has gained a little traction recently uses the spelling given by King John, ‘Liverpul’, and suggests Anglo-Norman roots meaning the springtime harbour. This breaks down the word to come from ‘Li’ as ‘The’ , ‘Ver’ as ‘Springtime’ and ‘Pul’, short for Pulvenarium meaning means anchorage. Liverpul as ‘The Spring Harbour’. It’s a very interesting theory as Liverpool has a massive tidal range, and would indeed have proved a lovely base for ships during Spring/Autumn, but been impractical the rest of the year, which could also explain why no permanent settlement had been set up there. It was until ship design improved (and the docks were built) that the town really exploded.