Yes - pass the -rdynamic flag to the linker. It will cause the linker to put in the link tables the name of all the none static functions in your code, not just the exported ones.
The price you pay is a very slightly longer startup time of your program. For small to medium programs you wont notice it. What you get is that backtrace() is able to give you the name of all the none static functions in your back trace.
However - BEWARE: there are several gotchas you need to be aware of:
backtrace_symbols allocates memory from malloc. If you got into a SIGSEGV due to malloc arena corruption (quite common) you will double fault here and never see your back trace.
Depending on the platform this runs on (e.g. x86), the address/function name of the exact function where you crashed will be replaced in place on the stack with the return address of the signal handler. You need to get the right EIP of the crashed function from the signal handler parameters for those platforms.
syslog is not an async signal safe function. It might take a lock internally and if that lock is taken when the crash occurred (because you crashed in the middle of another call to syslog) you have a dead lock
If you want to learn all the gory details, check out this video of me giving a talk about it at OLS: http://free-electrons.com/pub/video/2008/ols/ols2008-gilad-ben-yossef-fault-handlers.ogg
