Well, you HAVE to use the egg yolk, or you won't get mayo. I'm not in a position to look up Sally Fallon's recipe right now, but I can give you a general run down of MY mayo (and mayo in general), which we like better than Hellman's...
Egg - whole egg or egg yolk only is a bit of personal preference. Either will work. I find yolk only easier to work with.
Acid - try different acids, rather than sticking to the same one. They really do effect the taste. My favorite is 50/50 lemon juice and white wine vinegar - it's a really balanced combination, and doesn't interfere with flavor. All one or the other works fine, but the flavor isn't quite as nice, IMO. ACV, white vinegar, and red wine are all too harsh for this IMO. Rice or coconut vinegar would probably do nicely also, although I haven't tried them.
Emulsifier - this is usually mustard. I use a fancy French dijon - it's the only thing I use it for, so a small jar lasts a year or so. Again, the taste of this can effect the final product. The one I use is a Green Peppercorn Dijon.
Oil - this is your biggest taste point. The oil you use is going to dictate the over-arching flavor of your mayo. I can't stand olive oil mayo, and I love olive oil. Coconut oil doesn't work well because it gets too hard. I use organic cold-pressed sunflower seed oil. It is a very neutral flavored oil. I've also been known to use duck fat or bacon grease for part of the oil (just heat it enough to melt it, you don't want it hot). You have to leave the mayo sitting out a few minutes before using it, but the flavor of the fat really comes through, and both are spreadable at room temp (unlike CO).
Now, I know that Sally Fallon also calls for whey. I don't use it. It will make your mayo last longer, but I'm fine with making/using mayo within a week. I only make it maybe once a month, since we don't use it that often any more. Much past a week, I'm not comfortable with the mayo, but in the fridge for a week is just fine.
I would really suggest starting by tweaking the basic recipe and deciding what it is that you don't like/do like. Once you have that nailed down, it's going to be a lot easier to find a recipe your family will be happy with.
This is the recipe I use - although I make it in a food processor, not by hand. http://www.epicurious.com/recipes/food/views/Mayonnaise-241083