98 Grand Caravan - Intermittent Starting Problems

I need some advise on what may be causing this problem. On several occasions recently in normal weather conditions my wife's van has failed to start. This can happen after driving to the store etc. Wait a while and it wil generally start. Actually on about half the times this happens, it does start up but runs for only 2-3 seconds on other occcasions nothing happens - no clicks etc. You can tell when this going to happen because when you turn the ignition key,the electronic mileage data is not lit or is merely a line of dashes and the headliner computer info (miles to empty etc) is blank. Two tows to the dealer - first time they couldn't get it to repeat the problem- resulted in a diagnosis of a shorted battery which was replaced. However while that worked for a week, the problem is back. Logically it relates to something electric or electronic but its intermittent occurance makes diagnosis difficult so I would appreciate any info or suggestions as to what it could be. One further clue, when the AAA guy tried to start it (before dealer replaced the battery), he said it drained half the power from his battery box - however the engine wouldn't turn over at all.