The daughterboard is the graphics card. The GD610/GD620 is a quite common chipset for LCD/VGA graphics in laptops. It uses two 64k x 16Bit RAM chips to obtain 64k x 32Bit, which is the usual VGA memory (256 kBytes, but the VGA needs 32-Bit access to get the data fast enough to the screen). Those RAM chips have an access time of 100ns.
The chips on your mainboard are extremely common chips: They are 1M x 4Bit chips (with a total capacity of 4 MBit), and four of them make a bank of 1M x 16 Bit (as this is a 386SX system, the memory needs to be 16 Bits wide). It is very likely that soldering a second set of these chips (which can be stolen from any 2-chip or 3-chip 1MB SIMM) will expand the RAM to 4MB without any further need of customization. You should also add the decoupling caps (the big one is an electrolytic 1µF cap, labelled 105 on the other chips, the small one is most likely a multilayer ceramic cap of 100nF, which is the typical value used for them) next to the four RAM chips. The RAM chips already on the board have an access time of 80ns, so the chips you add should not be slower. They don't have to be by a specific manufacturer, though. All FPM (fast page mode) 1M x 4Bit chips with the correct package are compatible.
Thanks to Alex Hajnal for noticing that there is an electrolytic cap per RAM chip. It feels a bit over-the-top though. One electrolytic per bank and a ceramic cap per chip ought be really good enough.