Hi,
I'm sorry to have lost the camera you lent me. I'll buy a new one to make it up for you.
... to make up for you.
... to compensate for you.
Do all of the above sound right? If not, how should I reword them?
Say I'll buy a new one to make up for it. It refers to the camera, or perhaps to the act of losing the camera.
Another related idiom is I'll buy a new one to make it up to you. It here refers to the difficulty I have caused you. This idiom is not used much in situations whare I simply replace the item. Instead, it suggests that I do something that causes you a problem, and then try to do something else to compensate.
eg I'm sorry I lost your camera. I'll buy you an expensive dinner to make up for it.
Best wishes, Clive