Personally, I would go for the 8x56 over a 4x anything!
It is all very well telling someone to learn to shoot with a fixed power scope in the 4-6 mag range so as to learn steady free hand shooting, but then surely there should have been a comment regarding what maximum range a free hand shot should be taken at with said scope as this appears to be a novice asking? I could count on one hand how many deer I have shot free hand. Therefore, I don't practice it because I have realised through my experience there is almost always a tree, grass tussock, fence post, wall, bi-pod, shooting stick available to shoot from. Yes, an 8x scope will magnify the shake more but free hand at up to 50m will it make that much difference? I would rather teach someone what different objects can do to the POI when leaned on and how to address that than see a shot taken free hand. My eldest son is at that age where he wants to shoot his first deer and I will be drumming it into him the importance of a good lean, not learning to shoot free hand I'm afraid.
The first scope I ever bought for myself some 25years ago was a 8x56 Zeiss. I sold it when I moved on the .22-250 it was on and have regretted it to this day. Superb at last/first light and for the shot beyond the already mentioned 150m. I agree about the importance of range estimation however.
Each to their own but IMHO 4x fine for woodland or really close work but 6x or better still 8x would be the one I would go for. I can see however why some of the traditionalists on here would want to put a 4x on a 'classic' rifle.