As far as liquid oils are concerned, you mainly have a choice of oleic or linoleic oils.
Olive is an oleic oil, and you would have to use another one of those for a true replacement. Probably the best alternative with its good price and unparalleled shelf life is high-oleic sunflower oil. You could use it with a slight recipe adjustment.
Linoleic oils like soybean or regular linoleic sunflower oil are not replacements because they have VERY different properties. In larger quantities they make your soap drying, short-lived and prone to rancidity.
RBO is a strange duck that combines the qualities of some different oil types, but it contains too much linoleic acid to be a 100% replacement for olive oil. You can consider using a limited percentage of it.
Flaxseed oil is in a completely different (linolenic) category of oils that simply shouldn't go into soap. Those oils are for certain nutritional purposes, or otherwise to make oil paints or varnish. But not soap.
That would depend on how much olive oil you'd like to replace in your recipes. Flax seed oil and rice bran oil are much higher in the more sensitive/less shelf-stable fatty acids such as linoleic and linolenic than is olive oil. If you use too much of them in your recipe, your soap's potential for developing DOS increases. What is the % of olive oil you would like to replace? If it's really high, a better alternative (if you ask me) is High Oleic (HO) Sunflower Oil.
Simply put, DOS stands for Dreaded Orange Spots. Basically, if you use too much of the lesser shelf-stable oils (the ones with a high percent of linoleic and /or linolenic acids in them), your soap has a higher potential to go rancid. You can tell if your soap is going rancid when orange spots start to appear on your soap.