I'd have added a comment rather than an answer because I don't know php, but cannot do that yet. I'll speak generally from experience with other languages. The point of initializing a variable is that you want it to have a default value, unless otherwise specified. If there isn't a useful default value from the outset and you'd rather indicate that no value has been assigned yet, then you use some value consistently.
One advantage that I see in choosing null over "" is that an empty string is still a valid string, whereas null clearly tells us that no value has been assigned.
Setting the value to FALSE sounds wrong to me, since that would signal people reading your code that the variable is a boolean, rather than a string.
Whatever you choose, remember that in the end it's about how you use the variable. That is, if you intend to have some error handling code, then choose your values accordingly. Don't just assume that because you've initialized the variable, everything will be alright. Always think about all the scenarios and code paths and what would happen if the default (initialized) value were to be used.
Finally, if you're working in a team, check your company or team guidelines. It's so much better when everyone uses the same practice, even if not everybody agrees with everything!