The centre is small enough that the real choice is noise, price and which side of the Garonne you wake up on.
Triangle d'Or and the old centre — walk everywhere
The grid between the Grand Théâtre, Place Gambetta and Place de la Bourse puts almost everything within walking distance, including the quays. It is also the most expensive place to sleep, and the streets around Place du Palais stay loud until the bars empty.
Chartrons — quieter, still walkable
The old wine-merchant quarter north of the centre along the river: lower buildings, small wine bars on rue Notre-Dame, Tram B to La Cité du Vin. Roughly 20 to 25 minutes on foot to Place de la Bourse, so you trade a little distance for calmer nights.
Saint-Michel and the Capucins — cheapest inside the centre
The area around the Basilique Saint-Michel and the Marché des Capucins is the most mixed part of central Bordeaux and where beds inside the old town cost least. The market is working from six in the morning, so it is not a quiet base.
Around Saint-Jean station — practical rather than pretty
Hostels and chain hotels cluster by the station where the TGV from Paris lands and trams C and D start. About 25 minutes on foot or ten by tram to the centre, and it puts you next to the Quai de Paludate clubs whether you want that or not.