Hi Donna,
No, you don't have to buy a new heads immediately. You can start out with the heads that are already on the snare. Certianly, if they look like they are still in decent shape. I can see that the batter head is a Remo head of some sort, but I can't quite make out what it says. (I'm guessing that it's a single ply Coated Ambassador.)
When your son comes over to take a look at it, you should work together on assessing the different parts of the snare. Things like the condition of the snare wires, the smoothness of throw-off mechanism, the condition of the mylar straps that hold the snare wires in place, the existing batter and snare heads, etc.. You already have somewhat of a head start, in that you've probably visually assessed some of these areas when you were cleaning up the drum. Now is the time to take a closer look, and make sure that everything is in functional order. By doing so, you can help eliminate some of the headaches that can occur when you go to tune your snare drum, and can't seem to figure out why something doesn't sound right.
When it comes to tuning your new snare, you can start by finding a tuning that: A) Fits within the drum's range, and B) Something that you both find sonically pleasing. Now, if you need to add some damping to the snare, to tame some of the overtones that are inherent to snare drums, O-rings are a perfectly acceptable means of doing so.
If you can't find a sound that fits the criteria above, it might be time to look for a new set of heads. A new set of heads are great for giving you a "clean slate" to work with, not just sonically, but also in the fact that they should hopefully be in pristine condition physically.
So, give the current heads a shot and see what you come up with. If they need to be replaced, there are multitudes of choices available to help you achieve the sound you are looking for. Remember, there's nothing wrong with using some sort of damping on your drums (O-rings, Moongels, Snareweights, tea towels, etc...), especially if it helps you find the sound you are looking for.