If Borax is good at raising pH would lye (caustic soda) also be effective?
Is there a reason that lye is not being promoted but borax is?

Soda ash is fine for raising pH, but...it will increase the TA as well. If you need to just increase the pH, then Borax is the better choice since it will not increase the TA appreciably.

Maybe because borax is a bit safer to handle than lye?

Is borax easier to get than lye as well?

Soda ash is sodium carbonate. It can be found on the laundry aisle. Its called Arm and Hammer washing soda. Same thing as "pH up" at the pool store.

Caustic soda, or lye is sodium hydroxide. Yes, it's nasty. It's made from adding lime to sodium carbonate (although there areother ways to make it).

Correct. Lye is tricky to handle safely and not always easy to find. Borax is safer, easier to find, and does the same thing (though you need about five times as much to have the same effect).

If handling lye doesn't bother you (it probably should) you can use lye to raise the PH.

You can use caustic soda / lye if you want to, but as noted above, it's nasty stuff though so is Muriatic Acid. You can get it at The Chemistry Store though it's probably cheaper at places that sell to people making soap. The Pool Calculator lists lye in the lower section on the "Effects of adding chemicals".

Basically, 1 pound (16 ounces weight) of 20 Mule Team Borax (sodium tetraborate decahydrate) is equivalent to 12.2 ounces weight of Proteam Supreme (sodium tetraborate pentahydrate) which is equivalent to 8.0 ounces weight of pH Up / soda ash / washing soda / sodium carbonate which is equivalent to 3.2 ounces weight of caustic soda / lye / sodium hydroxide. It's concentrated stuff and, like Borax, will increase TA by only half as much as pH Up / soda ash / washing soda / sodium carbonate.

I've always been curious as to what situation would prompt an immediate need to raise pH in the first place. If a person has been following the BBB method, what would cause a downward drift of pH that wouldn't correct itself over time? (Excluding an accidental overdose of MA of course.)

You carefully dismissed the two most common reasons for raising PH: when you are using trichlor, and accidentally lowering your PH below 6.8. Ignoring those two, the next one would probably be mysteriously low PH on opening, which is rare, but happens now and then.

The accidental overdose is easy to do when you start with high TA. Mine was around 280 initially, and I had gotten used to dropping pH by about .2 with a half gallon of MA. When my TA finally got reasonable, it was hard to get used to adding much smaller amounts of acid.

Actually, I incidentally dismissed only one of the above reasons. (See the parenthetical statement at the end of my post.)

And the only reason I dismissed the first is because I normally don't think of trichlor when I think of BBB. My bad... But still it seems like it would take more than occasional use of trichlor to cause a problem...

Just tested the pH, after applying all the "AlkInc" and its nearly the same around 6.8...what a waste! Found a 4lb box of borax and added it, I will get more today!

Will order the test kit as well!

I would expect "AlkInc" to raise the alkalinity (TA) and only have a very small effect on the PH.