Is it a bad idea to have a home gym on the first floor of a house with a squat rack and barbell? Most home gyms with racks/barbells I have seen are in the basement or garage on concrete. My concern would be the occasional dropped deadlift or bailed clean smashing through the floor. Could mats, plywood platforms, and/or bumper plates mitigate serious damage?
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It definitely depends on the construction of your house. Some home features require extra care, such as grand pianos and hot tubs. Essentially, anything that puts a large amount of weight in a relatively small area.
Covering the whole workout area with another layer of 5/8" plywood will help provide additional rigidity and disperse the weight better. However, if you have an older house, you might have floor joists that are fairly wide apart such as ever 24". If that's the case, you'll need to add some more floor joists for extra support.
Now, if we are talking a total of 300lbs of weights, you should be fine. If you are pulling four big plates a side, you should definitely have the home inspected to see if it can handle the weight. This is true even particularly if you get a proper lifting platform (2" cement base with 2 layers of 3/4" plywood on top). The weight of the platform itself is enough to warrant caution.
To be absolutely sure, if you have a concrete foundation below you, you should be OK.
How much weight are you talking about? A good test is if you bounce on the floor do you go through it?
Seriously, I would recommend a plywood platform to keep your floor from getting messed up, make sure you put padding under it and that it's thick/strong enough for your weight plus the rack plus the max weight you plan on lifting. Make sure that any drops won't 'rattle' dishes, pictures, etc. in adjoining rooms. I think the main reason for most home gyms in basesments is due to the need to keep the wife happy OR general space issues. Here's my workout area (note the cat litter pans adding to the decor)