What cat does Barney refer to during your first meeting with Kleiner in Half life 2?

You all have missed the answer completely. It is simply a reference to a scene in the movie "the fly". The lab is even designed to be similar to the movie.

It's a joke about Freeman getting ready to teleport in the teleporter kliener built...its not a mention of a real event in HL you have to assume the cat did not fair well from the teleporter testing...think of the teleporter scene from galaxy quest when the pig creature was beamed aboard inside out then exploded.

This cutout copy from the half life wiki page about Kleiner might reveal something:

Barney worked as a double agent for the local Civil Protection forces at this time, while in the meantime ensuring that Kleiner's Lab would go undiscovered and safe from the Combine. At one point, Kleiner finished his teleport, and tested it with a cat, apparently resulting in its death, forcing Kleiner to resume work on it.

The first part says that Barney provided protection for Kleiner's lab, so he would know about it. Skip a bit about his headcrab and this is what it says:

When Alyx grew older, Kleiner also worked with her and Barney to rescue Citizens from the Combine, and evacuate them through the Underground Railroad located in the City 17 Canals to Black Mesa East. At the time of Half-Life 2, Kleiner is finishing up his teleport.

So it was before the events of Half-Life 2 but after the events of Half-Life. This explains why no one can figure out when it happened. It also was while Alyx was a kid, so it was not too long after Half-Life, because she says that her dad used to work with you in Black Mesa.

So, it happened a while after Half-Life. Wow, that was a long answer.

The "cat" is a reference to Kleiners pet Headcrab, which Barney has a phobia of. It could also be as discussed above, as an actual cat in an accident with the teleporter.

The cat they are talking about is Schrodinger's cat. Its an old thought experiment proving the Uncertainty Principle in Quantum Mechanics. The experiment goes: "if you place a cat in a box with a poison to released based on particle decay, you can not state what the cat is - he is neither dead nor alive until the box is open". It also shows the superposition theory where an unobserved item contains both states (the cat being neither dead or alive until observed). The achievements a pun based off this where the cat died so they observed that it is in fact dead and not both. https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat

Prof. Kliener has a pet headcrab that gets up to lots of mischeif. (Makes the teleporter go haywire, in HL2EP2 in the missile silo he climbs inside the machinery inside a hatch in the rocket and just sits there.)

I think that they are referring to Kliener's Headcrab. (Weird.)