Joy Of Pickling has a recipe for pickled lemons or limes - whole limes immersed in a brine of 1 T sea salt or canning salt per each cup of water (p. 76). Store in the fridge and ready to eat in about 3 weeks. She suggests using a brine bag to keep then under the surface of the brine.

AFAIK that is the only approved recipe but given their acidity I don't think safety is a big concern.

I think the problem with trying to can lime syrup is the same as with canning all citrus - heat making it bitter. But it should freeze well.

Dave