In addition to the other answers (particularly by Lekakis), some string replacements can also be used in the option --log-file=
as elaborated in the Valgrind's user manual.
Four replacements were available at the time of writing:
%p
: Prints the current process ID
valgrind --log-file="myFile-%p.dat" <application-name>
%n
: Prints file sequence number unique for the current process
valgrind --log-file="myFile-%p-%n.dat" <application-name>
%q{ENV}
: Prints contents of the environment variable ENV
valgrind --log-file="myFile-%q{HOME}.dat" <application-name>
%%
: Prints %
valgrind --log-file="myFile-%%.dat" <application-name>