Most cemeteries nowadays require all work to be done by one of their approved monumental masons, so you should contact the office to discuss whatever you have in mind. Even the wording of the inscription usually has to be approved.
Before you begin, you need to know who owns the plot (or the lease to burial rights in it), and there may be fees to pay to transfer it to you if necessary. There will originally have been a property deed, which should have been passed on through the normal inheritance process, though often they are lost over time. Again, the office may be able to tell you where to start.
Caveat: Procedures vary, and I know nothing of Scottish law. Some cemeteries, however, take a pragmatic approach and are quite happy for just about anyone to take responsibility for repairing a grave monument as long as it's done properly.