Hi Pam-my recollection, after perusing charts and lists in various guidebooks some time ago, was that August was generally the rainiest month, overall. Several locals we spoke with seemed to agree, but we didn't discuss precipitation patterns in much detail. However, glancing back now in a Lonely Planet book, which shows average rainfall by month in millimeters, not days, it says July and August are tops for Edinburgh (175-180 mm) with September a bit drier (155 mm). However, for Glasgow, August thru January is the wet period (over 100 mm each month, with August a bit wetter than September), while February thru July are drier (generally 75 mm or less, and April is the driest month of all, although July is much wetter than April). For Fort William and also over in Stornoway on Lewis, however, August is wetter then spring or early summer, September is wetter then August, and it really gets rainy from October thru January. Inverness, on the other hand, is definitely rainiest in August, then drier in September, then rainy again October thru January, before the spring "dry" season, if you believe those Aussies at LP. I also now see that Rick's Great Britain book suggests that, for Edinburgh, both August and September have the exact same number of rainy days (16), but no indication of how hard it will rain or how long the rain will last on a given day.
So, my general sense had been that it appeared Scotland got a bunch of rain in August, which tapered off in the fall - clearly now that's not true for some parts of Scotland! Sorry if I misstated things. Of course, when a hurricane moves from the Caribbean to the eastern coast of the US, then parks itself over the UK, it gets really wet no matter what some average weather chart says.
But rereading Toms' original question, he asked about temp. not moisture, so I was "all wet" from the outset. :-) Lonely Planet indicates (generally) September highs in the mid 50's and lows in the mid 40's. Rick suggests maybe a few degrees warmer (60 degree Farenheit highs for Edinburgh in Sept.).
Bottom line: prepare for rain and savor any dry moments, and it won't be too hot nor too cold, but keep a jacket handy.