Denbigh's Welsh name, Dinbych, is a compound of two old Welsh elements: din, meaning fort or stronghold — related to the word dinas, still used in modern Welsh for 'city' or 'fort' — and bych, an older form of bach, meaning small. Put together, Dinbych translates as 'little fortress', a fittingly literal description of a hilltop settlement that had been fortified long before Edward I's castle was ever built.
The English name Denbigh is simply an anglicised rendering of the same Welsh original, following a pattern common across North Wales, where English place-names are often adapted phonetic versions of older Welsh forms rather than separate English inventions.
The name's meaning ties directly into Denbigh's own history: this was, from its earliest recorded existence, a defended high point above the Vale of Clwyd — a 'little fortress' that grew, in time, into the walled medieval town and later market town that stands on the site today.
The Welsh name Dinbych combines din, meaning fort or fortified place, with bych, a form of bach meaning 'little' — giving 'little fortress', a fittingly modest name for a settlement that would go on to be dominated by one of Edward I's mightiest castles. The name almost certainly predates the current castle, referring to an earlier Welsh stronghold on or near the same commanding hilltop site above the Vale of Clwyd.
Confusingly for visitors, there is another Denbigh in Wales's placename history to be aware of: Tenby, on the Pembrokeshire coast, shares a related etymology (Dinbych-y-pysgod, 'little fort of the fish'), though the two towns are otherwise entirely unconnected and lie at opposite ends of the country.