I played 3 games of aram on a row with Jax. I don't know if it random but I don't think the odds of grabbing Jax out of my 90ish pool of champions three times on a row aren't really high. Plus I have never gotten a urgot or a wukong and just recently got my first trundle (which has been on my pool for about 4 months now) so yeah, they aren't completely random. What champs you play affects it slightly.
I would guess the champions you most play get a debuff on the chances (something like your top 10 most played champions get a 10% or 20% reduced chanced to being selected) as well as the champions you never play (my black list of champions you bought but never really learned how to play) so you end up with the champions that you more or less know its mechanics but you don't play it much for a or b reason.
When aram first started I mostly gots annies on my rolls, by that time the long range mids were the trend so annie didn't get much of play time even though she is on of my favorite champs. Now that assasins mids are the trend I get to play her more on SR, I haven't seen her on aram on a while.