what is the meaning of the hindi word " saun"?

"Saun" is not a Hindi word, but Punjabi. In Hindi it is called 'Shapat' and in Urdu 'Qasam'.

It's like taking up an oath or swearing. "Teri saun" means 'swearing by you'.

saun(h) is used as a short form of saugandh even in hindi speaking rural areas. Meaning is of course as explained by P'quaint!.