I am researching the origins of a 'Benjamin Browne' who married in
London in 1733 to a French Huguenot girl (Elizabeth Bertrand) when he
signed his name (twice) as 'Benjmain Browne'.  The name 'Benjamin' is
today frequently misspelled as ' Benjmain' due to a typing error but
in 1733, when written as such, it is legitimate.  My question is: does
anyone in France know of any examples (today or in the past) of their
spelling of the name - ie 'Benjamain' - being shortened to 'Benjmain'
- as though the middle 'a' is diminished or dropped - as 'Benj-main'
when pronounced in French??

As a french, I don't think Benjamin would easily became benjmain. the
middle A is quite strong and benjmain looks hard to be pronounced for
a french speaker.

I searched a database about given and family names with records dating
back to 1890. The database deals only with french data.
Benjmain is neither a family nor given name. On the contrary, Benjamin
is quite a common given name and not such an uncommon familiy name.
Only 5 persons were born during the 20th century under the Benjamain
family name, compared to more 1500 "Benjamin" and none with "Benjmain"

By a rapid look on Google results "Benjmain" is not uncommon but does
not link to webpages from France (rather west indies or Quebec)

Hope this help
François