As mentioned you need to carefully follow the initialisation
procedure and assume nothing about the state of the LCD, it may
have survived the brownout and initially just have the nibble
phasing wrong.  The triple command sequence to force it into 8 bit
mode is used to reestablish the nibble sequence in 4 bit mode, this
is part of the recommended procedure.

If you think that the LCD is getting stuck due to the sloppy way
the supply dips you may have to find a spare PIC pin and power the
display from the PIC.  This is perfectly OK and can be used to save
power by removing the 3 to 10 mA that the display draws if you have
nothing to display when the unit is sleeping.  Otherwise you can use
it to reset the display during periodic program excecution phases
when you want to force all your I/O devices into known states.  Be a
shame for the display to get the OK message stuck on when you are
busy scrolling the FIRE-FIRE alarm message and nothing happens.
Reading from the display is a good idea in critical applications
when you need to know if the controller is there at least.

You could also perhaps use a 3 pin reset generator IC to power the
display if you find one that will source the 3 to 10 mA that a
typical display requires.  This will generate a clean and fast
supply switch on (if you don't have any caps on the LCD side)
that will survive most any brown out.

Cheers
--
Kalle Pihlajasaari   kalle@ip.co.za   http://www.ip.co.za/ip
Interface Products   P O Box 15775, DOORNFONTEIN, 2028, South Africa
+ 27 (11) 402-7750   Fax: 402-7751    http://www.ip.co.za/people/kalle

DonTronics, Silicon Studio and Wirz Electronics uP Product Dealer