I have noticed more 83 year old men celebrating their second bar mitzvahs, and cannot find any source for this. Is it a new custom? Does it have any basis in pre-20th century practice?
The rationale behind it is that Tehillim describes a lifetime as seventy years in the verse
ימי שנותינו בהם שבעים שנה ואם בגבורות שמונים שנה
(90:10.) Thus -- the reasoning goes -- 83 is 13 years into your "second lifetime" which is as good an excuse for a kiddush as any. I do not know of any source for it prior to the twentieth century or of any book of halacha or official minhagim that quotes it, but that doesn't mean anything.