P4 motherboards use the 12V line for CPU power, just like modern ones. Any good modern PSU should work here, an old 300W PSU might if good design and no bad caps.

So: if you want to use the existing PSU check the caps. If they are bad, but PSU is a good design (Antec was great but caps were awful), replace the caps. If the PSU is cheapo crap, it's not worth restoring. Go for any good new one. Consider that the rating isn't the most important factor, bad brands advertise max peak power instead of sustained, and some just invent numbers. Brand is more relevant, weight is too (if you have two PSUs with same rating, the heavier is the better).

Ferenghi Rule of Acquisition # 322 ... A PSU that weighs only the same as the box it came in is worth the box.

BitWrangler wrote on 2021-09-16, 17:24:

Ferenghi Rule of Acquisition # 322 ... A PSU that weighs only the same as the box it came in is worth the box.

I once got an 8088 PC clone made by a Peruvian company.
It was using a large transformer and linear regulation on the secondary (no SPS).
Now I regret that I didn't keep it, not even take photos, because that thing was so overly heavy.
It was very exotic for sure.
Back then it must have been far cheaper to use a big Peru-made transformer than to import from Taiwan.
I do not remember what was heavier, the sturdy metal case which was possibly also Peru-made, or the PSU.
So I have no idea what this PSU was worth 😀

To the OP:
Use a PSU that generates 5 and 3.3V from 12V via DC-DC converter. These do not have the dangerous crossloading problem.

I suppose (but I leave the advices to the PSU experts) that for such system a modern new PSU should work fine. Using an old PSU is always a difficult choice cause most ones if time correct might work just like might be at the edge of their lifetime. They could work who knows maybe another decade or maybe they might stop working tomorrow. Often it might be a capacitor problem other times some other component why not. Also in the last years PSU improved a lot in things that in the past weren't exactly great. For example I've opened a faulty but good one Enermax P4 oriented 375W 24pin psu lately and very heavy with big heatsink, many protections for those times, fan control, active PFC etc.. it didn't work well cause strangely voltages went down even with few wattage load and I didn't understand why. Capacitors were all ok and good brand too, but who knows what is going to happen.
Anyway opening the PSU and looking at the back of the PCB there's an awful amount of soldering on all the circuit that surprised me for such good PSU consindering it was used on some industrial Pentium 4. Nowdays the back of the high end PSU has very clean PCB layouts, many protections, etc.etc.. so beside any PSU even new might give problems, it's always a difficult choice but I begin to think that after the warranty time expire, any PSU should be changed or at least cleaned, checked and saving money for the next one.

Thank you for all those answers!
I just checked the PSU I have and I think it's a cheap one, it's a Premier LC-B300ATX... It's probably better to take a new one.
What do you suggest? For exemple, I saw the Corsair SF450 or even the CX450, is that good enough? Or maybe you have some other brand to suggest?

Thanks !