As I too had been recounted many times that the term "Hindu" originated from the word "sindhu", I google searched "Is Hindu a corrupted form of Sindhu" and the search revealed predominant support for that theory, of course with a few articles demurring.
https://www.google.co.in/webhp?ei=j...rrupted+form+of+sindhu&newwindow=1&start=10&*
But this is hardly surprising because most of the articles relied on that premise which was popularised during Moghul and British rule. It is very similar to the now almost discredited theory of Aryan Invasion. Though the theory is discredited, the same continues to be taught.
The presence of alphabet for "sin" in arabic or persian does not mean much in my humble opinion, because scripts for writing were formed much later than native dialects and many alphabets have appeared and disappeared during the currency of a language. For example, the letter "La" in Hindi which was taught in schools when I was studying does not find a mention now and the same is the case with tamizh "ja". Raji mam's name would be written in tamizh as "rAsi" in the present day.
Further, I have worked in Gujarat for a number of years and it is very common especially for rural folks to pronounce 'sa" as "ha". Even one of the biggest cities in Gujarat named "Surat" would be pronounced as "Hoorat" by many rural folks and tribals especially in the tribal belts of Dangs, Central Gujarat districts of Panchamahals and also the people of Jamnagar district almost bordering Pakistan.
I think Hindukush mountain range may hold a clue for terming the inhabitants on this (Indian) side of the range as being called Hindus.