How to Update PS2 BIOS
You do not update a PS2 BIOS. It is a fixed file that never changed and has no versions to upgrade to. What you keep up to date is the emulator, not the BIOS.
Some guides promise a "latest" BIOS or a version to upgrade to. That is not real. This page explains why, and what to actually do when you want better compatibility.
Does the PS2 BIOS get updates?
No. There were no firmware updates for the PS2, and a dumped BIOS never changes.
The BIOS is fixed in the console and stays exactly the same forever. There is no patch, no newer edition, and no version to move up to. Any site that promises an "upgrade" is guessing, or trying to get a click.
Why people think it needs updating
They mix up the BIOS with the emulator.
The emulator, whether PCSX2 or NetherSX2, updates often, and those updates really do improve how games run. So it is easy to assume the BIOS updates in the same way. It does not. Every improvement you notice comes from the emulator, while the BIOS sits still.
What to update instead: the emulator
Keep your emulator current. That is where the real compatibility gains come from.
Update PCSX2 or your Android app to the newest build, and your existing BIOS keeps working with no changes. You do not re-add or replace the BIOS when the emulator updates. If you want the desktop details, see PS2 BIOS for PCSX2, and for phones see PS2 BIOS for Android.
What actually improves PS2 compatibility
The emulator, your settings, and sometimes a different region BIOS.
So the real question behind how to update PS2 BIOS is how to update your emulator. Keep it on the newest build for the widest game support. Beyond that, per-game graphics settings fix most of what is left, and for a region-locked title, switching to the matching region BIOS can be the answer. None of that is a BIOS update, but together they cover almost everything.
An "updated" or "latest" BIOS is a warning sign
If a site offers an updated BIOS, be careful. There is no such thing.
A file labeled "latest" or "updated" is either mislabeled or bait, because the BIOS has only ever had one form per console. Check any BIOS by its checksum before trusting it, the way the PS2 BIOS files explained page shows. Stick to a known-good file and ignore the version talk.
Getting a different region is not an update
Switching to another region's BIOS is not an upgrade. It is just a different file.
If a region-locked game needs another region, add that BIOS, but do not think of it as "newer." A Japanese or PAL BIOS is not an update to the USA one. It is a separate file for a separate set of games, as which PS2 BIOS to use explains.
Do real consoles ever get a BIOS reflash?
Rarely, and only during hardware repair. It has nothing to do with emulation.
On a physical PS2, the BIOS lives on a chip, and reflashing it is an advanced repair job. Emulator users never do this. For emulation you only ever load the dumped file, so this is not something you need to think about.
If your BIOS stopped working
The file did not expire. It is missing, moved, or corrupted.
A BIOS cannot go out of date, so if it stops being detected, something else changed. Re-select it in your emulator, or verify it by checksum and replace it if it fails. The BIOS not found fix page covers the usual causes.
Should you ever replace your BIOS file?
Only if it is corrupted or you need a different region.
There is no reason to swap a working BIOS for a "newer" one, because none exists. Replace it only when a checksum check fails, or when a specific game needs another region. Otherwise, the file you have is the file you keep.
Keep a backup of your BIOS
Once you have a verified BIOS, save a copy somewhere safe.
Because the file never changes, one good copy lasts forever. Keep a backup on a drive or in cloud storage, so you never have to dump or hunt for it again. If you reinstall your emulator later, you just point it back at the same file.