Yes, you can connect to an Oracle 8i database with the 10g client, but the 8i Database requires the 8.1.7.3 patchset, which you can get from Oracle's Metalink support site (requires login).
Here's an Oracle forum post with the details.
If updating your Oracle Database isn't an option, then you can have 2 different clients installed (in different "Oracle Homes" (or directories), and use the
selecthome.bat
file to switch between your installed clients.
For example, before connecting to 8i, you'd run:
C:\Oracle\Client1_8i\bin\selecthome.bat
or this to use your Oracle 10g client:
C:\Oracle\Client2_10g\bin\selecthome.bat