Geneva has roughly 200,000 residents and behaves like a city three times the size, and the reason is the river. The Rhône leaves the lake through the middle of town and cuts the centre in two, so almost every decision about where to sleep here turns out to be a decision about which side of that water you want to wake up on. The right bank holds the main railway station, the row of grand nineteenth-century houses strung along the lakefront, and the diplomatic quarter. The left bank holds the Old Town on its hill, the shopping streets below it, the Jet d'Eau, and the neighbourhoods where Genevans actually live. The two sides are about ten minutes apart on foot across the Mont-Blanc bridge, which ought to make the whole question academic. It does not, because the banks charge different money for different things, and because a Geneva evening feels entirely different depending on which one you walk home through.
Here is how the areas compare, and what each of three budgets buys in each of them.
Your room comes with the city's transport, and that changes the maths
Before anything else: every guest staying in a registered Geneva hotel, hostel or short-term rental receives a Geneva Transport Card, valid for the full length of the stay, covering the canton's trams and buses, the regional trains and the yellow Mouettes shuttles that cross the harbour. It costs nothing extra — it is funded by the tourist tax already sitting on your bill — and Geneva Tourism issues it as a digital pass by email in the days before you arrive, so it is worth making sure the address on your booking is one you actually read. Coverage is zone 10 of the Unireso network, which is the canton of Geneva minus one outlying village, and it includes the train in from the airport.
The consequence is simple, and most visitors miss it. In a city where a single tram ticket is not cheap, the difference between a hotel in the dead centre and one four tram stops out is nothing in transport cost and can be a hundred francs a night in room rate. The centre is compact enough to walk end to end in well under half an hour anyway. So choose for the quality of the street you come home to, not for proximity to a monument, and treat anywhere inside the canton as central.
One exception matters, and we will come back to it: stay on the French side of the border and the card does not apply.