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I have an old Pentium II 450 working now as my mother's computer, but
I'd like to upgrade it in the future to something several times more powerful,
and I'm wondering if there's a site where they simply measure how fast
each CPU going back to the old 386s in raw computational speed....

There isn't a single measure of power for CPUs, but Tom's Hardware has done the Mother of All Benchmarks, which includes over a hundred processors over 33 seperate tests..

The article

Results start here.

Entire listing of spec_cpu2000 benchmarks. However, the meaningfulness of these numbers depend on the meaningfulness of spec_cpu2000.

Same for spec_cpu 95.

there's simply so many different things that affect cpu speed, notably system config, and chipset architecture, that there isn't really any hard and fast measurement. best to just start doing some research into how they work and sort of make a best guestimate from there

Kinda more recent but theres also this