I'm not clear on what you mean by "stream" movies?
I have a very similar setup, an overclocked 1.58 GHz G4 Mini w/10.4.11 & 1GB RAM attached to 42" Samsung LCD TV attached to an 802.11n Time Machine. I can "stream" wirelessly movies that are stored on my G5 as MOV, MPEG, or other common video file formats over 802.11n wireless and they play fine on the TV played by the G4 Mini. I also have cable modem internet which is connected to the Time Machine and directly wired to the Mini over gigabit ethernet, but I still can't get smooth video of most "streamed" internet video.
I think it's likely you're talking about "streaming" as in "Flash video content" that is nearly ubiquitous on the web? If this is the case, about all you can do is hope and pray that Adobe optimizes Flash for better Mac performance. I can't see this happening any time soon for our now abandoned PPC hardware, so the best you can do is make sure you have the latest Flash plugin version 10.1.53.64 (check your version at: <http://www.adobe.com/software/flash/about/>). I don't think the G4 Mini will ever be powerful enough for streaming Flash video, which is one reason it's not on the iPhone or iPad. Any PC will play Flash video fine, so consider using a cheapie PC for streaming Flash if you absolutely need it. On the Mac side, even my dual 2.3 G5 struggles with Flash, so you'd likely need something very powerful for smooth streaming Flash on an LCD TV at full screen resolutions.
As I've harped before, Leopard is a 15-20% drag on any PPC Mac compared to Tiger, so you've already got a drag on the System. Second, if this truly a "first generation" Mini (1.25 GHz) it's probably not fast enough to begin with, my 1.58 GHz is "just barely" fast enough, and I'm ready to upgrade ASAP because it's often using 100% CPU to display video at 1680x1050. I'd advise you to downgrade to Tiger if possible to get maximum horsepower out of a G4 Mini (warning: Tiger & Leopard offer DIFFERENT combinations of resolution/refresh rates on my G4 Mini w/32MB Radeon 9200, so you may not be able to get the correct resolution/refresh rate for your TV. I think it's really strange these change between Tiger & Leopard, but they do).
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