Seat lacing is rawhide, not leather. It hasn't been tanned and pumped full of chemicals or oils to make it soft and supple as leather is. If it was, it wouldn't work for uses like snowshoes or canoe seats. It needs to be able to shrink tightly and be stiff enough to pretty much stay that way. Then it needs to be stabilized to keep it from both soaking up moisture and stretching out and also from drying excessively, which would allow it to continue to shrink over time and eventually tear itself apart. Varnish is by far the best substance available for stabilizing it in this fashion. A good marine varnish with U.V. inhibitors is best since U.V. will deteriorate and weaken the rawhide, just as it will just about anything else that is left out in the sun.
The process of making rawhide is pretty simple and starts by carefully scraping away any grease, tissue or natural fats from the inside of the hide and the hair fron the outside. This is followed by a lot of rinsing and when finished there is very little oil left in the stuff. This allows it to shrink and harden up. Adding oil to "recondition" a substance where somebody worked hard to remove the oils and fats in the manufacturing process just doesn't make sense. It's essentially more like an attempt to turn rawhide into shoe leather - which is not what you want for this job.
Yes, it's dry - but it's supposed to be dry. If it's still sound, a little sanding, scraping or wire brushing to remove any flaking finish and regular varnishing is all that it needs. You don't want to put anything on it that softens it or makes it more prone to stretching (like oil will). Assuming that it hasn't deteriorated over time from wet rot, mildew, rodent or U.V. damage, the biggest problem with old rawhide lacing is weakening from not being protected and stabilized. Getting damp from periods of either water exposure or humidity and then being allowed to slowly dry-out are perfect conditions for it to just keep on shrinking - to the point of breaking the fibers. The addition of oil will not cure broken fibers, it just makes them soft and oily. Varnish might help a bit by gluing a few of them back together, but chances are, it's not going to be a long-term fix, either. Even if it still looks good, if the lacing is structurally shot, it's done. I'd probably thin the first coat of varnish with a fast evaporating solvent, like naptha and then follow up with two or three more coats of straight varnish. If that won't fix it, nothing will.
The alternative is re-lacing with new rawhide, which isn't particularly difficult and can actually be kind of fun. Start by making a good-sized drawing of the existing seat with an accurate map of the lacing pattern. It's all done with a continuous strand that's spliced from shorter lengths as you go. Find either the beginning or the end of the current lacing and follow the strand as you make marks or arrows on the drawn lacing map so that you can duplicate the pattern later. Save the drawing as there is always somebody who is trying to re-lace their seats and is hunting for a map, but there aren't a lot of them floating around.
You can buy either full-grain or split-grain, real rawhide lacing from the snowshoe companies. Full-grain is stronger and as long as you're doing the project, you might as well use the good stuff. Different widths and weights of lacing also have different levels of strength. Narrower, lighter weights need tighter, closer-spaced weaving patterns to provide adequate strength, so if you plan to duplicate the original pattern, you want to use the same lace width to maintain strength. Soak the rawhide overnight before you start lacing, but not more than about 24 hours maximum unless you want to smell something R E A L L Y bad (...dead cow to the max...don't ask me how I know this). Lace them and let them dry for a few days until the rawhide is tight and hard. The edges of the strips can be fairly abrasive and sharp. Your pants will last longer (I'm not kidding) if you lightly go over the dried strips with fine sandpaper to round the corners of the laces a bit and knock down any sharp edges. Then varnish the seats well and maintain the varnish.