It's not explained on the wikipedia page. So if, apparently, 'H' is not relevant, why is it part of the name? And if it is relevant, why is it not explained?
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:
The same is true for all the other restriction enzymes, so is EcoRI the first restriction endonuclease isolated from Escherichia coli strain R.