A nursery near me sells something they call a Brown Sugar Fig. I can't find anything by that name on either forum or even on Google. Has anyone ever heard of such a thing?

Might be a Celeste?

If you are in SC, most nurseries only carry two varieties, Celeste and Brown Turkey.  It appears that brown sugar fig may likely be a Brown Turkey fig.

Ask them where it came from.  Tell them it doesn't Google. 

I went over there and they were pretty clueless. The mother tree was already there when they bought the nursery, so they just made up the name. I ate about fifteen ripe figs from the mother tree, they were mostly really bland. One or two were alright, they tasted a little bit more fruity than my Celeste (which is very floral tasting and faintly peachy). I didn't get one. I also passed on one they called "Texas Everlasting."

They also sold some Petite Negra and Negronne. Considering that they had mislabeled one of their "Brown Sugar" figs (three-lobed and single-lobed leaves) as a Negronne, I don't think they were careful about keeping the Petite Negra and the Negronne separate. It's a very laid back nursery. In the end they were super cheap so I bought one of their Petite Negra and immediately relabeled it as UNK. 

Ive heard brown sugar referred to as a taste profile.  And super cheap is hard to pass up.

Oregon Prolific is brown sugar tasting. Just had one not to long ago and was very nice.