Short answer is this: You do not need to enable any debugs to have Traceback messages sent to syslog, provided the router can send those messages upon crashing (which does happen sometimes, thanks to modular software design). You should also ensure that you have BGP
neighbor x.x.x.x soft-reconfiguration turned off because that will cause the router to store BGP advertisements even if a route-map rejects them.
Longer answer is this: You say that a couple years ago the router ran out of memory because it was storing full internet routes. You say that you added a route-map to reject all but the default route. If the router has the
soft-reconfiguration feature enabled, then it was storing the routes twice--once in the memory for BGP, and once in the regular routing table. If this is so, then it's possible that the router is running out of memory a second time now (two years later) because there are more internet routes now than ever before, and the memory is filling up for a second time because of the routes that BGP is caching, even though the route-map rejected them.
soft-reconfiguration is not on, then we take a step back and look for other causes. Generally speaking, it would be fair to say that routers crash for mostly two reasons--software bugs and being out of memory. We assume that you're not hitting any software bugs, since you didn't mention any software updates, config changes, etc. Usually running out of memory is caused only by a few scenarios: 1) the router has less memory than the software image requires, 2) the memory got all filled up by something, or 3) the memory is failing.
Perhaps you've had the second scenario in the past with BGP, but in a case like your current one where there is little evidence to make any conclusions at all, the third scenario seems the most likely. As @user56700 mentions, you should look in the log for any
Traceback messages, and also look in the flash for any
crashinfo files. Once the router has crashed, you will not usually be able to look at the log, therefore getting syslog to work correctly, as you mentioned, is a great first step. Generally, a router can perform some final actions when it knows that it's crashing, like sending syslog messages, and writing a memory dump to flash, so you have a fair chance at catching a Traceback message, if one is generated.
Unfortunately, that router model is end-of-life, so getting Cisco to look at a
Traceback or a
crashinfo file is basically zero. Might be time to upgrade that router!