Athenians used the lot (i.e. random selection) as a part of their selection mechanisms for government positions. Has the lot been used in other places and times?

Yes, the lot was a widely used mechanism in the organisation of republican governments before the French revolution.

I don't have time to type out a really long answer, but here are some examples:

In ancient Rome, lottery was used to decide the 'prerogative tribe' (the 'tribe' or voting unit that would vote first); since voting only continued until a winner emerged, and the rest of the votes were left uncounted, voting first could swing an election. It also showed which way the wind was blowing. (In modern US terms, think of a US presidential race, where instead of Iowa and New Hampshire always being the first to vote in the presidential primary season, it would be determined by lot). Random lot was also used, in the later republic, for the election of priests, where 17 of the 35 tribes would be chosen by lot to vote.

The Republic of Venice had a complicated arrangement of random lot and election, used in the election of the Doge (head of state), but also for lesser magistrates. Other Italian city republics did likewise: in Florence, Sienna and Genoa, certain officials were chosen by lot (although the constitutional specifics were complex and varied over time, a recurring feature of the Florentine Republic was the selection of officials from lists of nominated candidates by drawing beans out of a bag).

We think of medieval and early modern Europe as a continent run by monarchies, and we wouldn't be far wrong, but in addition to republican independent city-states, there was also a thriving and substantially autonomous urban life in many parts of Europe, with cities governed in accordance with charters that enabled them, under the distant sovereignty the monarch, to largely govern their own local affairs. Many of these cities were, in effect, republican, and random lot was widely used to select officials. My favourite example is the East Anglian city of Great Yarmouth, which for four hundred years, until the Municipal Corporations Act of 1835 abolished the practice, chose its vital public officials - the Herring Testers - by random lot.

Up to the modern age, lot was seen as more democratic than election, since election tended to favour the rich, the well-connected, the well-known, whereas lot gave equal change to the poor and obscure. It is important to recognise, however, that very often lot was not used (as its modern advocate recommend) to choose citizens randomly on the basis of equality from amongst the whole population, but from a narrower section of people qualified for office. The aim was not to ensure 'equal representation', but to ensure distribution of power across the ruling elite and to prevent any one family accumulating power.

Moreover, random lot was often combined with election in interesting ways. The first of these was called the 'scrutiny': a group of candidates would be nominated by an electoral process, and then the final decision would be made by random lot. The second was called 'brevia': the selection of members of an electoral college by random lot, but with the final decision being made by election.

The lot died out as concepts of political legitimacy changed: from legitimacy based on ensuring equal access to office amongst the elite, to legitimacy based on the elite's mandate from, and responsibility to, the wider voting public.