No, they have no memory. I tried that. They go through 8 patterns, then off, then through the 8 patterns again. Pulling the batteries out and back in had no effect-- they go back to pattern 1 when the power comes back. We looked at them earlier and now I am looking for a data sheet. I think we've come up with a temporary fix which involves a potentiometer dimmer, so we could turn the dimmer to low/barely on while I find the correct setting, then turn the brightness up. Off to Radio Shack tomorrow!
This is an OK fix if it works, but I am still interested if anyone would like to try fixing up the remote control thing for me in the future. I'll post info from a data sheet as soon as I find it!
Thanks again for everyone's help!