Both verbs sound fine within this context:
- She made this painting for a friend. - She created it; she produced it.
- She did this painting for a friend. - She was working on it.
The verb "do" mostly focuses on the process and the action while "make" focuses mainly on the result. (Cambridge - Do or make?)
But as Google Ngram shows "make a painting" is more popular:
As J.R. commented you can use painted, sketched, created, produced, crafted, or drew instead. However, each of these words carries a different meaning (see their definitions).
If you are unsure which verb to choose make "painting" the verb:
- She painted this for a friend.