Appreciating that the original questioner has long since ceased to need an answer, but understanding that other people might want to look at this, I'll replicate the basic answer found in the comments, with a little more explanation.
Sometimes flipping active to passive (and vice versa) can be complicated, but in this case it is easy. It's a simple future indicative, nothing to complicate matters. As usual, you put the object of the active voice (your secret) in as the subject, use the auxiliary to be in the same tense as the original (will be), the original verb changes to past participle (kept) and throw in the 'by' clause with the original subject (by me).
Your secret will be kept by me
There's no reason to move to the progressive participle keeping; you would generally only do that if the original were in the progressive aspect, too.
Whoever set the test for some reason mutated the future indicative auxiliary will into would. That is incorrect, certainly in terms of normal native usage in British or American English. "Your secret would be kept by me" would normally be a subjunctive, accompanying a conditional clause, or perhaps in response to a "what if?" question. "And what if I told you?" "Your secret would be kept by me." It can also be used for the-future-in-the-past, such as "you did not know that your secret would be kept by me".
Option 1 is actually using a gerund phrase rather than a passive voice for to keep. It is a noun phrase, and is grammatically valid and all that, but it isn't passive voice. The gerund phrase is "keeping of your secret", and it is a noun phrase that represent the action that might be infinitively phrased as "to keep your secret" (the 'of' is optional in the gerund phrase, and not correct in the infinitive). It is a noun phrase describing an action, so it can the object of to do. This option does use the passive voice, however, so the gerund phrase is the subject, and it "will be done". So, it is in the passive voice, but it is not a straightforward rendering of the original active sentence into passive voice - it is over-elaborated into something that no-one would likely say.
You can tell option 2 and 4 aren't passive voice versions of the active original, because they still contain I. The first-person pronoun changes between subject and object (whether direct or indirect, or indeed involved in some sort of complement), and as the original has the subject I, the passive version should have me.
