A relative pronoun acting as both object and subject?
Solution 1:
[1] I don't want to do anything you think [ ___ might be good for me].
[2] This is something he said [ ___ was good for me].
No, it can't be both subject and object simultaneously.
Your examples are both non-WH relatives, and since the subordinator "that" is omitted, they are sometimes called 'bare' relatives.
In both cases the missing relativised element (which we can call 'R') is not object but is located within a content clause inside the relative clause, where it is functioning as a subject. The ___ notation indicates the covert presence of the R element.
In [1] R is subject of the embedded "might" clause functioning as complement of "think", and has "anything" as antecedent. We understand that anything might be good for me.
In [2] R is subject of the embedded "was" clause functioning as complement of "said" and has "something" as antecedent. We understand that "something" was good for me
Solution 2:
What is happening here is that the relative pronoun is in a nested clause, rather than in the 'topmost' relative clause. That is why it is difficult to analyse, and why it might behave in a way that is slightly different from what one may be used to.
A basic relative clause:
This is the man whom I saw.
Main clause: this is the man
Relative clause: whom I saw
Whom is the object of saw in the relative clause. In the main clause, the relative clause is no full constituent (it doesn't fulfil a role in the main clause except as an attribute to man).
If you extract the relative clause from the main clause and turn it into a second main clause, you get this:
This is the man. I saw him.
The relative pronoun is replaced with a personal pronoun. In a normal main clause, the object would come after the verb, as here; but, in a relative clause, the relative pronoun is normally moved to the beginning of the relative clause. That is why we had the man whom... in the original example. Without this move, which is standard in (most? all?) Indo-European languages, it would be difficult to see where the relative clause begins, which in turn makes a sentence harder to parse.
Now a nested example, but let's begin with two main clauses:
I saw the man. I think he was mad.
The pronoun he refers back to a constituent in the first clause (man), so that is what we will replace with that/who. Notice that he is the subject of the subordinate clause he was mad; this clause is subordinate to I think (it is nested within it).
*
I saw the man I think who was mad.
We have replaced the subject he with a relative pronoun who. But this doesn't sound right: we want the relative pronoun to move to the beginning of the entire clause that used to be the second main clause (but is now the upper relative clause). So we move who:
I saw the man who I think was mad.
So the relative pronoun is still subject, but it was moved out of the inner clause.
Now on to your examples.
I don't want to do anything you think might be good for me.
I can split this into two main clauses (there will be a slight change in meaning, which is unavoidable, because it was a restricting relative clause):
I don't want to do anything. You think this might be good for me.
Turning it into a relative clause:
*
I don't want to do anything you think which might be good for me.
I chose which, because that has several different functions, such as a conjunction, which would have made it more confusing. Now we move the relative pronoun to the left:
I don't want to do anything which you think might be good for me.
You can use which and that almost interchangeably in this example. One feature of that is that it can be omitted. You probably learned a rule that that can only be omitted when it is object; but, in this case, it is subject, and yet it can still be omitted. The rule should probably be fine-tuned.
If you replace which with that, you can omit it here, and you arrive at your original sentence.
I don't see any relation between you think and the relative pronoun: they don't interact, as far as I can tell, and so the pronoun is not the object of think.