Cat hair and dander is the worst (compared to dogs - except, maybe, some hounds) because it tends to be more oily - thus sticky. This is compounded if the user smokes when using the computer too. You almost need to scrape that stuff off with a putty knife!
My advice is for your friend to get a different case, one that has removable, washable air filters. Then be sure to check them regularly and clean as needed.
Until then, ensuring the case fans produce a small amount of "over" pressure in the case helps tremendously. Over pressure occurs when the case intake fans pull more air in than the exhaust fans (including the PSUs) push out. This over pressure forces all the intake air to flow in through those vents in front of the fans (where there is - hopefully - a filter) instead of in through every (unfiltered) crack, crevice, slot and port.
With "under" pressure, a slight vacuum is created and that causes air (and all the dust, hair and dander floating in it) to be pulled in through all those unfiltered openings.
I agree that adding a filter to a case not originally designed to support filters can create a heat build up problem. That said, there are aftermarket filters just for that purpose, as seen
here. However, heat would have to be monitored and it may be necessary to add another case fan to compensate for the added restriction. And of course, the filters would require regular inspections and cleaning as necessary.
I also agree to keep computers up off the floor. That said, just 3 - 6 inches up off the floor can make a world of difference. Elevating it a few inches can also protect the computer from wayward vacuum cleaners, swinging feet, and wet carpet.