That Omnibook 425 appears to have 4.8v NIMH for the battery, so using the 4x eneloop batteries is just fine and you should be able to charge those in the laptop.
If it was lithium, it gets more complicated since the early lithium packs hadn't yet standardised on 18650 li-ion cells. The Battery Management System (BMS) on the packs should be pretty simple though and will still function if the batteries are removed for replacement.
See this video for an idea of what to do with modern lithium packs (where cell balancing is important and the BMS can turn itself off forever if tripped the wrong way). https://www.youtube.com/watch?v=h9okj71V2eg
Old laptops are easier, their BMS won't lock out and with NIMH cells, you don't need to worry about cell balancing or fire hazards so much.
To fit the cells, whether NIMH or Li-Ion, it's best to use a spot welder. I've tried using those spring loaded battery box things, but I think the laptop's power draw is high enough to go over the load that those little steel wires can handle. I say that because the runtime from AA batteries in place of the original 4/3A (17670) nimh cells was abysmal. Maybe AA batteries can't cope with that load well?
Soldering doesn't usually work well with batteries, and the heat can damage the cells.
For the spot welding, I recently got a little battery spot welder called a BIFRC DH20 pro, which does seem to work well. That along with some nickel strips would let you make up fresh packs without damaging the cells. Practise on dead batteries first to get a feel for the weld power needed and how to build the pack.
I tried replacing the 18650 cells on my Thinkpad 600 the other day, it worked but the pack didn't fit together right anymore and one of the cells got damaged so I didn't test further. I'm planning to refurbish a Libretto 70CT battery pack soon 😀
