I've tried a few different homemade recipes for vegan mayonnaise before, and been disappointed with every one, so I stopped trying several years ago. But I went to buy some vegannaise, which is nearly delicious enough to eat out of the jar with a spoon, and here in the Midwest I found it to be nearly double the cost that it was in Seattle. So I went back on the hunt.
I didn't have to search long. I made the following recipe twice, once with the called for canola oil, another time with extra virgin olive oil. Both turned out creamy and tangy, though the extra virgin olive oil version tasted, um... well, very strong of extra virgin olive oil. So my recommendation for anyone who wants to try this recipe is, feel free to experiment with various oils and proteins (almond milk instead of soy milk seems like an obvious thing to try for people like my mother who can't have any soy) but if you are to make this out of an olive oil, make it out of an old whorey olive oil instead of an extra virgin.
Vegan Mayonnaise:
1/2 C soymilk
1 c + 2tbs canola oil
1/4 tsp agave
3/4 tsp kosher salt (I used French fleur de sel)
1/2 tbs lemon juice
the zest from 1/8 lemon
1/8 tsp dry mustard
Put all ingredients into a blender and blend until creamy and emulsified, which is roughly 10 seconds.
This is where I found it, which looks like an lovely little place: http://veganepicurean.blogspot.com/2009/05/homemade-veganaise.html
Dangit, what a happy guy.