Question: I am doing my first photo shoot of a birthday party for a friend. I wanted to know which lens would be the better option to use for this. 
I have a Canon 50mm f/1.8 prime lens as well as the standard 18-55mm IS kit lens and a 55-250mm IS tele-zoom lens. 
So far, just getting to know my camera I have mostly played around with the kit lens. 
Now I'm not sure which of the others would be a better option.

Answer: 
Assuming that your camera has an APS-C sensor (18-55mm IS kit lens tells me that it does), the 50mm f/1.8 is probably a good bet for shooting portraits. Typically a photographer wants to use something around 85mm-100mm (on a full frame sensor) for a nice sharp portrait. Since your camera is APS-C and probably multiplies the focal length by 1.6x, you're looking at about an 80mm equivalent from your 50mm lens which gets you close enough to the sweet spot for portraits. Also the Canon 50mm f/1.8 is way sharper than either of the other two lenses that you mentioned. 

Summary: Unless you're using a 5D or 1D, you'll be just fine with the 50mm lens for portraits. If you want more candid, far away shots, try the 55-250, but for sharp portraits, the 50 is your best bet.
I have the same 3 lenses that you mention with the Canon 550D and a Metz flash. I've found that the 18-55mm kit lens does quite well when stopped down to f/8 and used with the flash indoors (bounce flash in particular). Make sure that you have the portrait picture style preset in use if shooting JPEG to make the portraits look warm. The lens covers the ideal focal lengths for indoor portraits including group shots. You can also get by without flash by bumping up the ISO - 1600+ is usable on the 550D, especially if you plan to share online exclusively.

The 50mm f/1.8 prime is also quite useful for individual and small group portraits. You can use wide apertures to blur the background. It gives pretty sharp results without having to stop down, and you can do without a flash to get decent shutter speeds.

Since you are planning to shoot a birthday, make use of the burst\continuous shooting mode to capture the important moments like cake cutting. This is also handy for group photos to reduce the chance of capturing people with their eyes closed. The prime again helps here, as it can be used in burst mode without a flash.

Also, be careful when changing lenses during the party - good time to get unwanted particles into the camera\lens.