If you like your old wheels, you want to keep the bike as original as possible, or you're trying to save money (who isn't?), you could certainly grab a set of whatever tire model is still available in 27 x 1-1/4. Practically any recently manufactured tires--even Kendas--will give adequate performance and safety for casual riding.
If you're committed to more "serious" riding, by which I mean longer distances, steep terrain, or higher intensity workouts, you'd do well to replace your wheels with something in the modern 700C size. This will allow you to get some nice aluminum rims, which give much better braking performance than the steel that your bike came with. It will also open up a selection of tire choices about 100 times larger than you have now.
You can get a decent set of wheels for a couple hundred dollars.
700C has a bead set diameter 8mm smaller than nominal 27 inch wheels. This means your brake calipers will have to reach 4mm farther. In some cases, there is enough adjustment range for the existing brakes to still work, but often the calipers must be replaced. The good new is there are nice, modern calipers available for maybe 50-ish bucks apiece, from Tektro and others. If you go this route, you'll have the added benefit of your brake performance improving even further.
As if there are not already enough reasons to make the switch, you'll also free up additional space in the frame and fork for fatter tires and/or fenders, should you ever decide to install some.
This might sound like an expensive and complicated rabbit hole to go down, and it is if you simply wanted to get your old bike out and make it rideable again. There's nothing wrong with getting a $40 set of Kenda tires (or some much nicer and more expensive Paselas), throwing them on, and enjoying the bike for what it is. If you are hoping to take your performance to the next level, I'd recommend upgrading the wheels and brakes as well as tires.
By the way, here is a vast wealth of technical information from Sheldon Brown:
Sheldon's Tire Page