I recently had my IIci stop booting reliably; when I opened it up, I found that two of the plastic tabs on the SIMM slots had broken (presumably due to weakening and constant pressure from the SIMMs?), resulting in a SIMM no longer making reliable contact with the socket pins.
I was able to easily desolder most of the pins on the existing socket with a combination of a desoldering gun, but 3 of the pins were exceptionally stubborn. I tried raising the heat on the gun, but even that wasn't enough. In the end, I had to use a pair of cutters to dismantle (ie, destroy) the socket and expose the remaining pins on both sides, and then carefully use desoldering braid on the pins with a soldering iron set to slightly higher heat.
I was able to find dual-socket replacement SIMM sockets from Jameco; unfortunately, I discovered (*after* destroying the original socket) that the Jameco part doesn't match the original's spacing between the two SIMM sockets.
Rather than give up, I decided to take a hacksaw blade to the dual sockets; after separating them, I was able to solder them right in. You can see the cut edges here:
The machine is back to booting reliably with fully populated memory. I also now have a ton of extra 30 pin SIMM sockets with odd spacing (they were really cheap ...).