Casting lots ("sortition")
The most standard means of making a random selection was drawing lots (sortēs): everyone involved would put their names into a container, then one would be drawn out at random.
convenere viri deiectamque aerea sortem
accepit galea, et primus clamore secundo
Hyrtacidae ante omnis exit locus Hippocoontis
The men gathered around and a bronze helmet gathered the lot[s] that were tossed. And first out of all of them the place of Hippocoön Hyrtacides is drawn, with applause...
The Trojans here are deciding on an order for the archery contest: all the competitors put their names into a helmet, then they were drawn out one at a time. Idiomatically, sortes "lots" can refer to any random chance, or to the action of Fate.
Flashing fingers ("micatio", Italian "morra")
This was more equivalent to "rock-paper-scissors" or flipping a coin, a simple game used to resolve minor disputes when necessary. Two people would count off, as in rock-paper-scissors, then simultaneously hold up a number of fingers and guess what the total would be. (A modern variant involves one player winning on odd totals, the other on even, but I haven't found classical attestation of this version.)
This game was ubiquitous enough to make its way into a common proverb: dignus est, quicum in tenebris mices ("he is an upstanding person, with whom you could play morra in the dark"). Cicero also mentions it along with sortition in De Officiis 3.23: a matter can be decided randomly, quasi sorte, aut micando. (The proper term for this in Latin was digitos micare, "to flash the fingers", or often just micare.)