If you say 'I'll see what I do' you are not conveying the same meaning as that of the sentence 'I'll see what I can do'
'I'll see what I can do' means that you will see what is possible for you to do. Generally, you may use such sentence as a response to a person's request.
'I'll see what I do' may mean that you, as the speaker, will check your (perhaps habitual) actions.
Student or Learner