Regarding replacing the original TMS4108's like for like, then one major problem is that you need to get the right versions NL0 or NL1 suffix, to match the one being replaced. And if you've got 1 or more failures in both banks, then would have to source both types. Attached is a full datasheet on it, for reference, as can sometimes be difficult to find.
A couple of tricks I've read about for diagnosing faulty rather-similar 4116's in ZX Spectrums:
First to feel if some are getting hotter than others. And remove any that are getting much warmer.
You could also try checking the resistance of data pins to ground / each of the supplies, to check for any internal shorts in one (although if both banks are on the same bus, without isolation buffers, then there would be both ones in each bank on that particular data in/out line. But can't do on Address-lines, as all shared.
However, 'scoping the datalines and checking for anomalies maybe better.
The 2nd trick is to place a known-good DRAM IC on top of each of the board's DRAM IC's, and see if any differences occur. This is meant to work if IC is dead but not shorting. And probably works best if only one has failed.
Note: It's probably wise to power-down each time, before holding IC on top, and would either need to use the same TMS4108 suffix type or a 4116.
I've also seen Data Out etc. pins cut and then resoldered, to help isolate faults, but not too good for it looking fairly untouched.
I think I have tried these techniques once, but Ididn't have too much luck. So I found it easiest to just socket all 8, and put known-good ones in to start with. Then swap the originals back in a few at a time, to find which ones were faulty.
- Especially as I have acquired quite a few other the years very cheap from radio rallies - mostly new unused ones. Although this place list them at only 40p each (around what they eventually dropped to when new, before discontinuation made them retro-priced):
http://www.diverse.4mg.com/compfull.htm (If they still exist / have these, as website is rather old)
Being as the PET 6502 CPU only runs at 1MHz, compared to nearly 4MHz on Spectrum Z80A, then access time isn't as important. And so even the slowest (250 / 300ns?) ones should be OK.
There's quite a lot of different make 4116 datasgeets here, for finding out what the suffix numbers mean in terms of access times (some have higher single number suffix for fastest speed rating / lowest access-time, rather than it being the start of the access-time:
http://www.citylan.it/wiki/index.php/DRAM_16k_x_1
This also shows an interesting DIY ROM Board converter, to use a single EPROM:
http://www.themadhowes.org.uk/pet/index.html
Although they don't show a schematic or the underside layout of the veroboard.