I have taken a couple of cutting from mine, which is the black-leaved variety. I think I did it last summer or autumn (I have a feeling it was demonstrated on GW on TV at the time). I think I just pulled off a small shoot and cut its top growth back to two main leaves, then put it at the edge of a pot of compost with some perlite added for drainage. Both cutting have rooted, although I nearly lost one because the pigeons pulled off the new growth. I have now put old hanging-basket frames over them both for protection.
Elder is very strong and persistent. You can cut it right back to near the ground and it will shoot back up again in the spring/summer. I don't know how you'd get rid of it, but I am told it is lucky to have elder in your garden.