I have met people that this occurred to and the problem is often the proof of the family status. That is, unless there is absolute proof (on a halachik - legal - basis), many people in this circumstance will "convert out of doubt". If the original Jewish mother was too long ago (as in centuries), it is possible that one of the intervening generations may not have been Jewish, but the family history forgot that. Additionally, accepting someone as Jewish cannot be done unless the evidence can be accepted in Bais Din (Jewish religious court). The evidence involved needs to be discussed with a rabbi who is an expert in such matters.
That is why the people that I knew performed a fully valid conversion. If the family history was true, then they were Jewish and the "conversion" while not required was a sign of renewed commitment. If the family history was inaccurate and they were not Jewish then the conversion made them Jews.
Many Conversos (Jews forced to convert to Catholicism in 1492) actually went to the Mikvah and symbolically "converted" when they managed to escape to the Netherlands and resumed their Judaism. The Rambam (Maimonides) also discusses people who had been in similar circumstances. I do not have the reference to that as it is from memory of a shiur that I attended a while ago.