What does 'H' in BamHI stand for?
These enzymes are named by the bacteria and the strain from which they are isolated. For BamHI this is Bacillus amyloliquefaciens strain H. In the beginning the enzyme was named BamI, which was later changed to BamHI.

See these two references:


Isolation of a sequence-specific endonuclease (BamI) from Bacillus
amyloliquefaciens H.
Recognition sequence of specific endonuclease BamH.I from Bacillus
amyloliquefaciens H.


The same is true for all the other restriction enzymes, so is EcoRI the first restriction endonuclease isolated from Escherichia coli strain R.