I believe what you have here is a piece of a 'cut' coin, probably from a Spanish Reales, which circulated among the American colonies until about 1850 or so. The 'H' is a feature remnant of the original coin, probably 'Hispaniarum', Latin for the Spanish Empire, and the band with the small diamond shape is a remnant of a Spanish Crown often portrayed on their coins. If you can find a book of Spanish Colonial coins you might be able to isolate the pattern and denomination to identify exactly which coin this came from, and a close idea of the date it was minted.
When a trader didn't have small change he or the merchant would take a larger coin and cut it into appropriate sized pieces, hence 'pieces of eight'. The merchant might then put a mark of his own, verifying weight, metal content (ie silver in this case), and value, so it could continue to circulate in good faith. The 'M' and the incised arrow symbol on the opposite side are merchant's marks.
There are also counterstamps for when a coin already issued was re-weighed and assigned a new value by the government, but these look like small seals or raised relief letters, usually with a circle of dots around it. These counterstamps were used as an anti-counterfeiting measure, and to prevent undervalued coins resulting from clipping or filing by unscrupulous people who were sometimes beheaded when caught.
Sorry I don't have the resources readily available to help you further identify this piece, but this may give you a place to start. Good luck!
