That seems like a very unusual (and large) way to produce a pseudorandom sequence. Why not a linear congruential generator or LFSR? They have approximately the same state storage requirements, but much less static data.
This must be pretty fast (and compact) code: rndindex = (rndindex+1)&0xff; return rndtable[rndindex];
The example C code at https://en.wikipedia.org/wiki/Linear_feedback_shift_register looks like it would compile to more bytes, and require more CPU cycles to execute.
Based on https://en.wikipedia.org/wiki/Linear_congruential_generator it seems LCGs also micro-optimize for memory rather than CPU performance.
Doom was probably squeezing every last cycle out of 386 @ 25 MHz machines to render frames, but they did have at least 4MB–8MB RAM; 256 bytes seems a good trade-off.
This may also present the advantage of being tweakable to avoid repeated sequences that better generator may naturally produce but are not perceived as 'random enough' by humans.