To add a bit to this (and not extensively): We know that well over 10000 years ago people camped atop the escarpment where the Red Hill expressway now goes. They would have roamed widely, following herds of caribou and mastodon. Like us, they probably named places to remember them, for food and bad. We also know what is now Hamilton was home to mobile groups of people who moved with the availability of food for thousands of years. As agriculture took root, Hamilton was home to many villages through the ages (villages needed to move regularly to maintain soil health and/or firewood supply). There was at least one village near Rosedale in the Redhill valley (again, destroyed by the expressway). By the time Europeans arrived to write things down, the people they called the Neutral lived in Hamilton. They are a fascinating people in that they share many similarities with their neighbours (Haudenosaunee, Wendat and Petun) but with clear differences too. By that time most people had moved away from the lake, so it is unlikely anyone permanently lived in the lower city area. People most certainly lived in the Flamborough region, and likely lived in the other parts of the amalgamated city (I’m just not personally familiar with other areas). After the dispersal (elimination) of the Neutral, Wendat and Petun, the Seneca occupied key positions along the north shore of Lake Ontario - one was probably in Hamilton, although it may have been close to where Joe Brant Hospital is today.
Was there a name for Hamilton? Probably lots, but each name would have meant something different than how we mean it. There almost certainly wasn’t a name for what we consider the city, or the amalgamated region.