What do you do in your spare time?
What is the proper way if I want to convert above mentioned sentence into reported speech.
He wanted to know what I did in my spare time.
or
He wanted to know what I do in my spare time.
or
He wanted to know what did I do in my spare time.
The first, or the second.
The basic (present, positive) form is I do X in my spare time.
So the past is I did X in my spare time.
In a direct question (or a negation) you need the auxiliary do to invert:
What do you do in your spare time?
What did you do in your spare time?
But in an indirect question, there is no inversion, so no requirement for do support:
He asked what I do in my spare time.
He asked what I did in my spare time.
Both of these are fully grammatical and natural: the first implies that this is recent, because the expectation is that whatever I did when he asked, I still do it. The second does not have that implication: it could be referring to a past time. But it doesn't have to be - it could be recent as well.
Your third option is dubiously grammatical, but it is common in casual speech:
He asked what did I do in my spare time.
This does not obey the rule above that in indirect questions the subject and verb are not inverted. Nevertheless, people say it. I suspect that it is a blend of the quoted direct question He asked "What did you do in your spare time?" and the indirect He asked what I did in my spare time"
Your third example is clumsy whereas the first two are both concise and correct. Which one you prefer depends on the context.
If you imagine that your brother sees you speaking to a friend and is curious to know what the conversation was about, you would probably reply:
He wanted to know what I do in my spare time.
This implies that you continue to occupy yourself in the same way in your spare time. It also suggests that the conversation is recent.
or you might even reply:
He wants to know what I do in my spare time.
Implying that the questioner has just spoken to you.
However, if you were a witness in court and being asked about a conversation that took place between you and someone else several months earlier, you might well reply:
He wanted to know what I did in my spare time.
This does not raise the question of how you continue to occupy your spare time and it suggests that the conversation is not recent.
Which form of the verb you prefer depends on whether regard the question as relating to an earlier period (how you then spent your spare time) or to the present (how you continue to spend your spare time). It also depends on whether you regard the conversation as something that took place in the past or that is still fresh.
