Can I keep the same room?
'Stessa' — stress the first syllable: 'STES-sa'. Double S. 'Tenere' — 'te-NE-re'.
Ask when you want to stay in the same room rather than moving. Moving rooms is disruptive, especially if you have unpacked for a long stay.
'Tenere' = to keep/hold. 'La stessa camera' = 'the same room'. 'Stessa' (same, feminine) agrees with 'camera' (feminine noun). This is a natural follow-up to agreeing to extend the stay.
Devo cambiare camera se resto?
Do I have to change rooms if I stay?
Asks the question in reverse — confirms whether a room move is required
La mia camera è già riservata per altri ospiti?
Is my room already reserved for other guests?
Asks the reason why a room change might be needed
Preferisco non cambiare camera — è possibile?
I prefer not to change rooms — is it possible?
States a preference politely
Italian hotels will always try to accommodate a guest's preference for the same room. If unavoidable, the room change is typically offered at no extra charge and with porter assistance. In Italian hospitality, inconveniencing a guest unnecessarily is considered poor 'ospitalità'.