The reason why olive oil is so common in soap recipes is that it has historically been the the easiest high oleic oil for soapers to find. It is a good source of oleic acid, which helps to make a mild, lathery soap when combined with stearic and palmitic acids and a dab of lauric and myristic acids.
Nowadays it is easier to find other high oleic fats including avocado, rice bran oil, high oleic sunflower, HO safflower, HO canola, etc. Probably the closest to olive are any of the HO oils, with rice bran a close second, and avocado running third. These fats may also have the advantage of not being adulterated or faked, which is a widespread problem with olive oil.
To substitute any of these fats for olive, just substitute them in any recipe that calls for olive. (Edit: Be sure to run the changed recipe through your favorite
soap recipe calculator to make sure the NaOH weight is correct for the new blend of fats.)
You don't have to use any high oleic oil at all, if that's where you want to go. The classic "no olive" recipe is a lard soap, either 100% or with up to 20% coconut oil. If you look at the fatty acid profile for lard, you'll see it is a reasonably decent source of oleic acid. Not as good as the oils mentioned above, but not too bad.