In the Roman era both linen and wool were widely available so both would be appropriate for this project. Silk was also available but it was very much a ‘fancy’ fabric used to show off wealth and status as it had to come from abroad.
A lot of the fabrics found in military contexts like excavated forts are tightly woven twills that would be good at keeping the weather out. If you want to have a historically appropriate fabric then a wool twill would be a great choice.
Otherwise, any wool or linen in both plain and other weaves would be a good choice. The fabric weight depends on your local climate and what you intend to use the cloak for. However if you want to hand sew the Lunetius it can help to choose a slightly lighter weight fabric to make sewing easier on your hands.
If you are not working with any historicalness in mind Lunetius can be made out of a variety of fabrics. A woven fabric would drape in a different way compared to a stretch or knitted fabric but you are only limited by your imagination.
More information on Roman cloak fabrics can be found in:\ Jorgensen, L. B. (2004) A Matter of Material: Changes in Textiles from Roman Sites in Egypt’s Eastern Desert, in An Tard 11, pp 87-99