Please try this:
./configure --with-aolserver=/home/sam/downloads/aolserver.org/aolserver-4.0.10

Such change didn't work. AOLServer is actually configured and compiled with prefix of /usr/lib/aolserver4 (as per Debian module).

sam@caveman:~/downloads/php.net/php-5.1.1$ ./configure --with-aolserver=/home/sam/downloads/php.net/aolserver4-4.0.10/
loading cache ./config.cache
checking for Cygwin environment... no
checking for mingw32 environment... no
checking for egrep... grep -E
checking for a sed that does not truncate output... /bin/sed
checking host system type... i686-pc-linux-gnu
checking target system type... i686-pc-linux-gnu
checking for gcc... gcc
checking whether the C compiler (gcc  ) works... yes
checking whether the C compiler (gcc  ) is a cross-compiler... no
checking whether we are using GNU C... yes
checking whether gcc accepts -g... yes
checking whether gcc and cc understand -c and -o together... yes
checking how to run the C preprocessor... gcc -E
checking for AIX... no
checking whether ln -s works... yes
checking if compiler supports -R... no
checking if compiler supports -Wl,-rpath,... yes
checking for re2c... re2c
checking for re2c version... invalid
configure: warning: You will need re2c 0.98 or later if you want to regenerate PHP parsers.
checking for gawk... gawk
checking for bison... bison -y
checking for bison version... 1.875 (ok)
checking for flex... flex
checking for yywrap in -lfl... yes
checking lex output file root... lex.yy
checking whether yytext is a pointer... yes
checking for working const... yes
checking for flex version... invalid
configure: warning: flex versions supported for regeneration of the Zend/PHP parsers: 2.5.4  (found: 2.5.31).
checking whether to force non-PIC code in shared modules... yes
checking for pthreads_cflags... -pthread
checking for pthreads_lib...

Configuring SAPI modules
checking for AOLserver support... /home/sam/downloads/php.net/aolserver4-4.0.10/
configure: error: Please specify the path to the root of AOLserver using --with-aolserver=DIR

Just pass the configure option ONCE and with the right path.
The right path is the _INSTALL_ prefix of aolserver.

Thanks -- the instructions I had for configuring must have been out of date.

While configure now completes and make completes as well, this does not result in producing a shared module (e.g. libphp5.so) as expected.

Variations attempted:

./configure --enable-aolserver=/usr/lib/aolserver4 --enable-shared=aolserver

./configure --enable-aolserver=shared,/usr/lib/aolserver4

Both result in configure reporting:

checking whether the gcc linker (/usr/bin/ld) supports shared libraries... yes
checking dynamic linker characteristics... GNU/Linux ld.so
checking how to hardcode library paths into programs... immediate
checking whether stripping libraries is possible... yes
checking if libtool supports shared libraries... yes
checking whether to build shared libraries... no
checking whether to build static libraries... yes

The build produces the CLI executable but not the desired shared object.

The configure option is called "--with-aolserver"..
Please ask further support questions elsewhere as this is not a support channel and this problem you're having isn't a bug in PHP.

This turns out to have a root cause of the manner in which Debian 3.1 installs the aolserver4 package. By building the aolserver4 source package myself and installing the entire package into a single root (/home/sam/aolserver) I was able to get:

./configure --with-aolserver=/home/sam/aolserver

To successfully complete.