For my VX it recommends 10w30 for the manual trans...I am guessing 5w30 or 0w 30 wouldn't hurt it?
The thing to keep in mind, is that transmissions need different properties out of their oil, than car engines do. And even when the transmission was originally speced for regular 10w30 motor oil (as my Honda CRX was), it can still be a bad idea to use a normal 10w30 synthetic or even some other weight dino oil.
The main reason for this, is the synchronizers in the transmission are apparently very sensitive to exact friction properties, and can wear out early (or otherwise get damaged) if you don't have exactly the right amount of friction/feedback in your oil. And synthetic motor oil (which is designed to optimize the properties that help an engine, which are not alway the same properties you need in a transmission) has noticeably less friction/grab than normal dino oil (so will behave differently in a car's transmission, even if/when both are 10w30 oils).
So if you want to put synthetic oil in a transmission originally designed for 10w30 dino oil, buy a synthetic oil specifically designed for use in your transmission (with the proper add pack, to work well with the transmission's synchronizers). For example, both RedLine and Amsoil make a synthetic "Manual Transmission Lube" that is specifically designed to be used in such transmissions. I currently have the RedLine "MTL" ("Manual Transmission Lube") in my CRX, and it's amazing how well it works (the shifting is so smooth and easy, it's almost unreal). And the nice thing about the RedLine product, is that it's even rated to make a small difference in fuel economy (by lowering some of the power losses in your transmission).
However, don't try to be cheap, and save a few $$$ by using engine oil in your manual transmission (even if your transmission was originally speced for 10w30 engine oil). Instead, get a lube specifically designed for transmission use. I know that you may very well "get away with it", but an engine oil just won't work as well as a lube specifically designed for transmission use (I know, as I actually had Mobil-1 engine oil in my transmission for a time, and my car started shifting MUCH NICER after I switched to the RedLine MTL). And it's also "false economy" (to buy a slightly cheaper engine oil) as well, as I've heard reports that using the wrong oil (such as a synthetic engine oil) in the transmission, can also accelerate the wear on the transmission (and thereby cause the transmission to fail sooner). All things considered, the extra money on a good transmission lube (that matches the type of transmission it is to be used in), is well worth it IMHO....