I usually recommend laptops for media servers. They typically draw about 10-15 Watts while idle, possibly less if the screen is turned off. A modern i3 desktop system designed for low power draw is going to be up around 30 Watts. An older Core 2 Duo system around 60 Watts. And an old P4 will be over 100 Watts idle.
Media servers tend to be left on 24/7. Over a span of a year, a 50 Watt difference in idle power works out to 440 kWh. At the average U.S. price of $0.12/kWh, that's over $50 in extra electricity costs each year. If your system draws more power, or you're in a region with higher electricity prices like Europe, the cost difference is even bigger. And since you already have the laptop, might as well try it first, then if it's not satisfactory you can build a cheap computer or buy a newer laptop.
Unless you're planning to regularly serve multiple video streams, a 5400 RPM laptop drive bigger than about 250 GB should be more than fast enough (about 30-50 MB/s). Even an external drive plugged into USB 2.0 should be OK (25-30 MB/s). I don't know how fast the drive on an Inspiron 1300 is.
It'll be better if the laptop has a gigabit ethernet port, but it's not essential. A full 1080p raw blu-ray stream is about 18 MB/s, so won't fit on 100bT ethernet. But 720p, SD, and 1080p MP4 streams are much smaller and will fit in 100bT just fine. 100bT is more of an annoyance when you use the laptop as a file server, because reads/writes are limited to about 12 MB/s. But most video streams fall well below that cap.
Will the videos be stored in a format the smartphones can play natively? Or will it need to be transcoded in real-time? I suspect the CPU on the Inspiron 1300 won't be able to transcode HD streams in real-time. Even the AMD Brazos and some Atom CPUs have problems with that task. So that might be a reason to opt for a newer system.