The registers throughout my house are 4 inches by 12 inches. That's one of a few standard opening sizes for registers in the US. Mine are in cut-outs on the floors, which is a standard but not universal arrangement. As I mentioned earlier, I don't have convenient access to the ducts feeding the upstairs rooms, so I can only work with the registers and the box-like vent openings.
The cheapest sources of fans these days are standard sizes of PC cooling fans. A 92mm fan is just over 3.6 inches and would fit nicely in the 4 inch opening. And, as it happens, 3 of them side by side would be less than 11 inches long, fitting nicely in that dimension and allowing room for any other odds and ends I need to fit there. (Another popular vent opening size is 4 inches by 10 inches. Someone with those vents would only be able to put two of those fans side by side.) I'm not sure how I would mount them. It's easy or cheap enough to make or buy brackets that can join standard 92mm fans to each other. I think I would then fabricate either some legs to stand them up or some hangers to hold them up. That will probably need some experimentation. I want to make sure that whatever I use doesn't end up creating some kind of vibration noise when the fans are going.
Speaking of noise, most people already appreciate the general idea of trading off fan performance for fan noise. People putting together very low-noise PCs typically use larger fans running at lower RPMs. If I'm going to use off-the-shelf PC cooling fans that fit inside the vent boxes, 92mm is the starting point. There are 100mm fans available, and they would fit into 4 inches, so I might consider those. In either case, the size of the fan is going to be constant in the consideration of other fan parameters: air flow volume, static pressure rating, RPMs, and noise level.
The parameter least appreciated by the average person is the static pressure rating of a fan. Even though the manufacturer can give a maximum static pressure rating for a fan, it's more important to understand the trade-off between static pressure and air flow volume. This engineering note by Johann Tang gives a very easy to understand explanation of the relationship: https://blog.orientalmotor.com/fan-basics-air-flow-static-pressure-impedance. When static pressure reduces air flow volume, you can make up for some of that by increasing the RPMs of the fan, but that means increased power consumption (which you probably don't care about) and increased noise (which you probably do care about).
In the case of a register booster fan, there shouldn't be much obstruction above it to create a static pressure differential. The grill on the register shouldn't count for much. But there might be something to think about on the other side. For the individual rooms, the vents are fed by 6 inch round ducts. As far as I know, there are not any smaller duct segments in the paths. The 6 inch duct has a cross-sectional area of just over 28 square inches. A 92mm fan has a cross-sectional area of just over 10 square inches (a 100mm fan has a cross-sectional area just over 12 square inches). With 3 fans, that's a fan opening area of 30 to 36 square inches. The duct will not be able to deliver air at the same rate the fans can expel it, so that will create, effectively, a kind of "suction" on the back side of the fan, or a difference in static pressure. (At least, that's my theory. I hadn't really thought about much of this before a few days ago. I could be completely wrong.) I don't know how to calculate or measure that static pressure difference, and my plan is to just make up for it experimentally by boosting the fan RPMs.
Speaking of RPMs, you might already know that PC cooling fans come in 2 varieties these days. Long ago, PC fans just operated on 2 wires, power and ground. In today's 3-wire fans, there is tachometer-style feedback on that 3rd wire, so the motherboard can vary the supply voltage or other tricks until it gets the RPMs it wants. A technique called pulse width modulation (PWM) can be used to rapidly turn the fan on and off (like some kid rolling a tire down the road) to affect the speed. In a 4-wire fan, the PWM control is part of the fan itself, and the 4th wire is used for the motherboard to tell the fan how fast it should spin. The motherboard still monitors the tachometer output so that it can adjust the PWM signal until it sees the target RPMs it wants. This blog article by T.K.Hareendran does a good job of explaining the various signals involved: https://www.electroschematics.com/4-wire-pc-fan/. Now, in that description, just substitute "microcontroller" for "motherboard".
How loud are fans? If the manufacturer of a fan doesn't tell you how loud it is, it's probably safe to assume that it's pretty loud. Anybody who makes fans that are on the quiet side goes to a lot of trouble to let you know the details. You will sometimes see the sound rating in sones and sometimes in decibels. (This article has a good explanation of the relationship: https://learnmetrics.com/sones-to-db/). Because decibels are more commonly used for PC fans than sones, I convert any sones numbers to decibels to keep things straight. And, speaking of sones or decibels, it's pretty easy to find a million charts on the internet giving examples of various kinds of sounds. For example, it's common to see the sound of a whisper being about 30 db. One thing to keep in mind when evaluating fan noise is that there's a big psychological difference between hearing 30 db whisper for a second or two versus a 30 db fan noise that continues indefinitely.
That's about all I have for now. I'm evaluating (aka, window shopping) 92mm and 100mm fans. I'm also thinking a bit about the circuitry that would drive all this (probably an ESP32 and a little bit of glue circuitry).