Well, Rome was a Republic (res publica) for basically as long as it existed (Founded 753 BCE in tradition, lasting possibly until 1453 CE, depending how you want to define it), and obviously the rules of the senate changed from time to time in that period. I'm just going to go up through the late Republic, since I don't really do imperial stuff. Maybe /u/celebreth can tackle that.

In Roman tradition, as represented by Livy and Cicero, the senate was originally Romulus' 100 best buddies, which he had organized into a advisory council. That's obviously completely mythical, not least because no one thinks Romulus was a real person, and so couldn't organize anything.

In fact, we know almost nothing about how the senate was originally constituted. It seems clear from the phrase patres et conscripti that it included both patricians and plebeians. How the members were selected, though, is almost completely obscure.

Sometime in the 4th century, by the lex Ovinia, the censors were given the task of setting the membership in the senate. By the end of the 3rd century we find the custom developing that members be former magistrates, who meet a certain wealth requirement, without a serious moral problem in their background (determined by the censors, of course). Censors could also just appoint people to the senate as needed. In about 149 BCE, another law was passed that made former tribunes of the plebs senators (many would have been senators anyway, by holding the quaestorship previously). And in various points of crisis - notably Sulla's dictatorship - loads of people were just promoted into the senate. Sulla basically doubled the senate's size from 300 to 600.

What this means is that for much of the Republic, membership of the senate was in fact determined by the voting bodies that elected magistrates and tribunes - that is, the Centuriate Assembly, the Tribal Assembly, and the Plebeian Assembly. Who exactly was the demographic that controlled the votes in those assemblies is currently a matter of serious debate. I think Mouritsen, Morstein-Marx, and Yakobsen all have relatively recent contributes to that debate, with some (Yakobsen and Morstein-Marx) arguing for wider popular political authority. Traditionally, the Centuriate and the Tribal Assemblies are seen to be heavily dominated by the wealthy, due to the weighted votes of the class structure of the Centuriate Assembly, and the heavy distribution of the Tribes into the rural areas of Italy. The censors were also elected by the Centuriate Assembly, BTW, so whatever independence they had in managing the membership of the senate was in fact granted by one of the same bodies that elected magistrates.

However the argument about how much popular participation in the voting assemblies turns out, it seems clear that, while the senate was not selected by its own members, it was selected by the 1%.

Both Julius Caesar and Augustus also introduced large numbers of people into the senate without having won an election, and I don't really know how the institution continued in the imperial period. Tiberius put a stop to public election of magistrates, and instead those were indeed elections exclusive to the senate. The censorship fell out of use as an office. There must be other developments to senate membership in the imperial period, but that is outside my period.

OH EDIT: Whoops - forgot to add that this post is almost entirely drawn from the article on the senate in the OCD. That's where I got the dates and the technical stuff.