Yes, keep the name but apply it to something else. The original d was a denarius, a silver coin. Perhaps the 10p could become a 'penny'?
Yes, they could be just in sets for the time being - the denomination could therefore be 'one pence' and not a 'penny' as such. After all, a penny has been a silver denarius, a Saxon sceat, a small medieval silver coin, a Maundy denomination, a copper trade token, several types of copper coin, then a bronze, finally a (now worthless) steel decimal 'bronze'.
You do realise that the 1p is now worth less in real terms than the farthing was when demonetised in 1960?