Mine made EXACTLY that same sound, until I blew it up 15,000 miles later. I removed/replaced the clutch multiple times and am convinced it was the transmission. It started after I either put in my ACT XT, or swapped the differential, can't remember.
I have also ALWAYS had this noise with every car I own, some worse than others. Thankfully my Miata is fairly quiet compared to most of them. I always install correctly, using new parts, after cleaning all surfaces and applying grease, and yet I always end up with some kind of un-factory sounding TOB roar.
Swapped the clutch and tranny. Lube all the point indicated below, it's silent. I'm pretty sure the noise was the TOB rattling on the fork.
Any idea what cured it? I always assumed mine was related to rattles from the TOB on the fork, but never knew what could really be done about it. Always use new parts, and grease. Always have the noise anyway.
when I pulled the trans I took a look and the TOB; still looked in good shape and spun smooth. However, when sitting there on the input shaft there was a good .125-.250" of play on the fork. I'm pretty sure that particular noise was the TOB rattling on the fork since there was zero grease on the fingers of the fork or the slots on the TOB. It made a similar noise when I would jiggle it around.
When I put the new bearing on the fork after greasing both heavily, there was less apparently play and it made no noise since it was dampened by goo.
All my other noise are gone now too with this new trans so my drivetrain is finally quiet.