I converted my 2002 Trek 4500 hardtail MTB into a hybrid and it works really well as a commuter bike, though the 9.5 miles each way I go on it is a bit too much to be honest and I'll eventually switch over to something a bit nicer. I think you need to post some more details about your commute. Typically, switching to smooth-ish tires and a rigid fork are what people recommend to do first. They definitely helped me the most, though I would suggest not the lightest road tires but something designed for durability to avoid flats. As for everything else, I don't really see the reason and it gets expensive quickly. Just a decent set of tires and a new fork (you'll need to get a special suspension-adjusted fork to compensate for the lack of suspension) will eat up a good chunk of your budget. A decent set of wheels will eat up most of the rest. Sure, upgrading to better and lighter components might increase your speed slightly, but unless you have money to burn, switching out something just for weight reasons is probably not worth it.

Personally, I did end up upgrading a lot of components on my bike, but that was primarily due to parts wearing out. Not because I thought the slight loss in weight would help. It didn't make sense to do it before that and I gained very little from it.