This is an outgrowth of the discussion here. The rex was indeed elected (or perhaps acclaimed would be better term) by the people at the suggestion of the Senate, as far as I recall but I do not remember term limits - perhaps I am missing something?
UPDATE: Anixx has kindly provided the source for the eponymous hypothesis in his answer below. In my opinion, if that is all the evidence that can be adduced in favour of this hypothesis, then it can be very safely dismissed, for the following reasons:
It is not clear where did the number 8 come from to begin with. Why assume a term of 8 years and not 7 or 9? Since there are no references to ancient literary or epigraphic sources in the hypothesis, any guess seems to be just as good.
Rome had 7 kings whose reigns lasted 38, 42, 32, 24, 38, 44, 38 and 24 years plus minus 1 year, depending how one counts). Three of the numbers (24,24,32) fit the hypothesis mathematically but the others don't.
The theory that the term was 8 years and then somehow became 24 is just that: a theory. I think it fails Occam's razor.
The suggestion that Tarquinius Superbus, the last king, was ousted because his formal term of 24 years ran out flatly contradicts everything we know about the end of the kingdom and the establishment of the Republic - the king's ouster was a revolution, a highly irregular proceeding. So his rule of exactly 24 years was actually atypical and is in fact a point of evidence against the theory.
UPDATE #2: I googled a bit to see if Koptev's theory about term limits for the rex has gained any academic currency or has been suggested independently by other scholars - and it seems that it hasn't. For instance, A.A.Schiller in the book Roman Law: Mechanisms of Development explains about the different theories proposed about the extent of the rex's powers but he takes for granted that the position was for life. Here's a quote from p. 137:
There is as much speculation regarding the manner in which the rex was chosen as there is with respect to his powers, since the details must be reconstructed from later reminiscences and reflections of the Republican era. The post was for life, and the successor is traditionally said to have been selected by means of the institution known as interregnum.
UPDATE #3: Just to set the record straight, Koptev's theory has been published by him in a reputable academic journal: (http://www.jstor.org/stable/10.2307/4436788). However, it still seems to not have gained any acceptance or currency - Google Scholar shows just two citations so far, one in another work by Koptev and another in an Italian book. It's in footnote n.31 on p.66 and it begins "No comprendo los argumentos de A. Koptev", i.e. "I do not understand the arguments of A. Koptev".
'Three Brothers' at the Head of Archaic Rome: The King and His 'Consuls' Author(s): Alexandr KoptevReviewed work(s):Source: Historia: Zeitschrift für Alte Geschichte, Bd. 54, H. 4 (2005), pp. 382-423Published