Which one is correct in or on own way?
- I usually help my closest friends on/in my own way.
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In my own way is the idiomatically correct expression. I can't think of a situation where I would use on my own way.
Now, in my way and on my way are both valid expressions. You could say in my way instead of in my own way if you wanted to; the own just provides a bit of emphasis.
If I am on my way, I am proceeding to an expected destination. One more thing:
You are in my way.
I am in my own way.
In this case, in my way means interfering or preventing me from doing something, most typically moving from one place to another. If, for example, you are standing in a doorway and I'm trying to go through it, you're in my way.
Now, if I am in my own way, it means that I'm creating difficulties for myself in accomplishing a task. For example, if I'm trying to lose weight and won't stop eating, I am in my own way. (In this case it would be more common to say I'm getting in my own way.)
"In my way" refers to the definition of way as "preference", or "style", so it would be more appropriate here:
Frank Sinatra sang that he always did things in his own way.
"On my way" refers to the definition of way as "route" or "path".
On my way to school I saw my old friend Jim.
However, "in my way" can also refer to something obstructing your path:
The train was delayed because a herd of cows was in the way.
In your case, definitely "in my own way".
"in [my/your/her] own way" means "in a way that is special to that person". The only time that "on" and "way" are used together (that I can think of) is in "on [my/your/her/the] way" which means you are in the process of coming somewhere (e.g., "I am on my way to the store.").