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The difference between Giur, conversion, and Giur l’Chumra, conversion – to-be-sure, (or to be strict), is that in the case of regular conversion, the person converting was for sure not Jewish beforehand according to Jewish law. With Giur l’Chumra, there is a possibility that he/she is Jewish, but since there is doubt, he /she undergoes the process of conversion to be sure.

If the person already has a Jewish name, they may keep it, and if not, they should take a Jewish name.

If your question refers to an actual case, a proper Rabbinical Beit Din for conversion should be consulted. Only orthodox conversion is valid, any other kind only causes confusion.