I was talking to my neighbour some more about this, and he thinks it might be a "chicken and egg" issue with the Honda cooling fan relay. When he replaced the relay on his son's Accord, he also had to replace the battery - it tested completely done for. It could be that the cooling fan runs and drains the battery - and that fan would drain it easily, it draws way more current than something like a dome light. But, he wonders if a weak battery might cause the relay to degrade, then the dying relay drains the already weakening battery, and presto, you have a REALLY dead battery. He's seen some other instances of where a weak battery or alternator has caused issues, and thinks this might be one of them.