The FAQ for Hakchi has a list of mappers that should work in an NES Classic Edition's built-in emulator.
Common Nintendo-made discrete boards
0 (NROM): Thwaite
2 (UNROM): STREEMERZ
3 (CNROM): Solar Wars
7 (AOROM)
All Nintendo MMCs in their most common board configuration
1 (MMC1): Nova the Squirrel
4 (MMC3)
5 (MMC5)
9 (MMC2)
10 (MMC4)
Quick Disk images for Famicom Disk System
And a few third-party discrete boards
86 (JF-13)
87 (Jaleco VROM switch)
184 (Sunsoft-1)