A silly question since the character is probably rarely used:

Ser Jacelyn Bywater
Response: After Ser Jacelyn Bywater is killed, each opponent must choose and discard a card at random from his or her hand, if able.

This card was being used in a melee today... it seems contradictory to say each opponent must choose and discard a card at random, so are the cards discarded at random or is it player choice? Or is it supposed to mean each opponent discards two cards, one at random and one they choose, or is it just a poorly worded card that I am overthinking?

It's oddly worded, but not impossible to sort out. eg. Turn your own hand face down, shuffle it, then choose a card and discard it.

It's basically just discarding a card at random, except that the wording actually has a "Choose" (wording indicating a target, which could potentially interact with other cards).

I'm wondering if the "choose" means that the effect can be canceled by the likes of Maester Vyman. The card is chosen at random during initiation, but after it is chosen at random, it is turned over in order to be discarded. At that point, it is open knowledge if it is a Stark character, or a noble-crested character, and Vyman or KLE Eddard should be able to cancel the effect. If the word "choose" wasn't part of the effect, this cancel wouldn't be possible, since the effect wouldn't have a target.

You are really over-thinking it.

Is it not possible to choose something at random? Or to make a random choice?

If I were to tell you "choose a card at random," you'd know I only meant for you to pick one card - and that you had to use some method other than specific selection to come up with it, wouldn't you?

"Choose and discard a card at random."

"Choose and discard a random card."

"Choose a random card and discard it."

"Choose a card at random and discard it."

They all mean the same thing.

(And yes: ultimately, the word "choose" is used here to indicate that the random card is a target of Bywater's ability - with all that a "target card" entails.)

I wish I was less anal-retentive, because ktom's answer makes me want to do a card-text search and find out how "random discards" are worded in various iterations. :/