If my customer changes their mind about the dates, can I amend a hotel booking instead of cancelling and booking again?

Sometimes, but not always, and it depends on the supplier behind that rate. Some hotel suppliers accept an amendment through the hotel booking API, so you can send the new check in and check out dates against the same booking reference and your customer keeps the original confirmation. Others do not offer an amendment call at all, and the only way to move the stay is to cancel the existing booking and make a fresh one.

Even where an amendment is possible, the new dates still have to be available and they have to be priced again. The rate your customer paid was for those particular nights, so shifting the stay usually means the supplier recalculates the cost. If the new dates come out higher there will be a difference to collect, and if they come out lower, whether that difference comes back is set by the supplier rules rather than by you. A rate that was non refundable when you booked it normally stays non refundable, so a date change may be refused even when the room is sitting empty.

Cancel and rebook is the fallback, and it is only comfortable while the booking is still inside its free cancellation window. Outside that window you would carry the penalty on the old booking and pay the full price on the new one, which can cost your customer more than the change is worth.

Before you promise a date change, open the booking, see which supplier it came from and check what its cancellation deadline is, then try the amendment first and keep cancel and rebook as your second option.