I believe it was called "Dashing Dan's Diner" by those who survived eating there (I'm not one of them). Probably named that more because you had to make a mad dash for the latrine than for the LIRR "mascot."
Mike: I like your aviator BTW! I've posted this across a number of forums AND asked some retired LIRR/historical guys and nothing... Amazing: tens of thousands of employees over the years, fans, etc. No other photo responses, insight as to the food, years there, etc... I'm stumped..