Will an SSD help prevent laptop overheating?

In general, no, because SSD's offer no significant advantage in terms of power consumption over conventional, mechanical hard drives (see the bottom of this post for an example), especially when you compare this difference to the power consumption of the system as a whole.

What it truly comes down to is how much power each drive consumes under a load, or at idle. Why do you care?

Well, both drives are like a closed system, and from the first law of thermodynamics, the heat we put into the system must be equal to the heat that comes out (since no work is actually done aside from moving the platters, and even then the motion of the disk platters, eventually through microscopic and macroscopic processes dissipate into heat as well by the second law of thermodynamics). Long story short, if the SSD draws more power, it dissipates more heat.

Unless you find a solid state drive that draws less power then your current hard drive (or any drive, for that matter), keep what you've got ;)

Just to give some numbers to my claims, for example, an OCZ Agility 3 SSD uses 1.5W at idle and 2.7W under load, whereas a 1TB WD Scorpio Blue HDD uses only 1.4W under load and a mere 0.6W at idle!

Not likely. The majority of the heat is probably being generated by the CPU and the discrete GPU (an ATI HD 5650). This is common on laptops because it's relatively easy for the fans and vents to become clogged up with dust and dirt. It might also be caused by poorly-applied thermal paste, or a heatsink may have come loose.

I would check the vents for obvious blockage. Open up the case and clean it if you can. If that doesn't help, or if you'd rather not open the case, then contact the retailer or the manufacturer and report the problem - it should still be under warranty after all.

But still feel free to upgrade to an SSD, because the performance improvement will be awesome...

usually while playing games but sometimes while watching videos or using Skype video calls for a long time.

All activities that are either streaming video or involve intense CPU/GPU processor usage. Check for extremely poor air-flow, dust and fluff caught in heat sinks or obstructing air flow past your video processor, central processor and memory cards.

Sticking an SSD drive into this machine will probably be like putting a chromed radiator cap on a plugged up radiator. Purty with bragging rights, but the heat dissipation still stinks.

Well, since you asked about other suggestions. (presumably besides his one of changing the hdd to ssd), and I can include pictures of the other suggestions here.

You can get things like this. laptop fan coolers. laptop cooling pads.