Cylindrical tenons in drilled mortises are a very old form of joinery that is strong and simple to make. Making the tenons the correct size is critical for the success of the joint.
Depending on your design, choose a suitable diameter for your cylindrical mortise and tenon. For chair or stool legs, 1" is typical. For stretchers, 1/2" - 5/8".
To get the best fit possible, make a sizing gauge using the *exact same bit* and *exact same wood species* as you plan to use for the mortise. Not all 1" drill bits are precisely the same, and a hole in oak may behave differently than one in pine.
Strike a line, drill a hole centered on the line, then carefully bandsaw the hole in half. Voila, you have a sizing gauge.
On the lathe, mark the tenon's length (extra long for now). Set a pair of outside calipers to a bit larger than your desired finished size and turn to that dimension.
Whenever possible, orient the work so that the tenon is on the tailstock side of the lathe. That way you can remove and check the workpiece, then return to the lathe for adjustments.
Use the sizing gauge to bring the tenon down to final size. Be sneaky! Take light passes and check frequently. Make sure the tenon has parallel sides--no taper, bumps, or hollows.
This step is not necessary, but it can be helpful. Cut a small chamfer on the end of the tenon to help it get started in the mortise.
Final checks! The tenon should fit into the mortise with no play or wobble. It may require light hammer taps, but should not need heavy blows.
If the tenon is too big, return it to the lathe. Avoid using the disc and belt sanders--you are nearly guaranteed to lose concentricity and roundness, as well as introducing taper. If you already cut the ends off the workpiece, it may be possible to use the 4-jaw chuck on the lathe (ask for instruction).