Okay, I'm ready to give this a go
According to
https://www.parkytowers.me.uk/thin/hp/t5540/
Quote
RAM: The RAM is a standard 200-pin SODIMM. The one fitted is manufactured by hynix and is marked: PC2-6400S-666-13-C0, 512MB, DDR2, 800, CL6. The part number is MT4HTF6464HY-800E1. The VX800 data sheet says that it... "Supports up to two 64-bit DDR2 667 DIMMs (4 GB max)" from which I assume that, with a single DIMM socket, the maximum amount of memory supported is 2GB. In December 2017 Samir emailed me to confirm that a 2GB part does work.
Does this mean that this should work?
1GB DDR2-667MHz PC2-5300S 200Pin 1.8V SO-DIMM RAM Laptop Memory For Hynix