Attribute efficiently learning the relevant attributes of juntas with membership queries
Juntas can be tested in an attribute efficient manner given a membership query oracle: http://www.cs.cmu.edu/~eblais/papers/TestingJuntas.pdf Testing is an easier problem than learning, and the attribute efficient result is relatively recent. This might be a good place to start poking around. 

Incidentally, the hard part about learning juntas is just identifying the set of $k$ relevant variables. Once that is done, you can learn the junta through exhaustive querying with $2^k$ queries (which might be "attribute efficient" since it is independent of $n$. In any case, you can't hope to do much better since the function can be arbitrary on these $k$ bits...). The testing algorithm linked to seems to work by trying to identify relevant variables, so it might solve your problem if you go through the details.