Note: Almost all the following information is gathered from the Finnish Iron Age Facebook page. I am not an expert and can be totally wrong about any of this. 😛
Wire is categorized into soft, half hard, and full hard. Soft metal brends more easily but does not hold its shape as well. Full hard is great for holding sharp corners and such but it is more brittle and harder to work with. For Finnish coil decoration it is recommended to get half hard wire, which is in the middle. It is stiff enough to hold the shape, but soft enough that it isn’t brittle and is easy to work with. Soft also works, since the coiling process (being worked) hardens it.
The exact metal composition of the bronze doesn’t seem to be that important, as they didn’t have the technology to do exact mixes anyways. But note that there IS a difference between bronze-colored art wire (probably made of aluminum,with a colored outer layer), and actual bronze wire made of bronze. The “art” wire is much lighter, and any sort of scratching on it will show the silver-colored aluminum underneath the colored layer. It’s probably still fine generally (not at all period, though), but bronze or brass wire isn’t that expensive considering how much time and effort you’ll be putting in to this, so you might as well get the real thing, IMO.
I am saying “bronze” wire, because that is how it is referred to, but in actuality it was more brass like. The period wire DID have tin in it. Most replicas I have seen look more brass like, and from personal experience bronze coils get very unshiny brown (like an old penny) VERY quickly.
The wire they used is smaller than you would expect. The absolute biggest wire was 1.5mm (about 15 gauge) that was used as single rings (not coils) on the edge of a shawl (not an apron). For aprons, you should probably go with 0.5 – 0.7mm (20-24 gauge) for the interior decoration, which was done in the later period. The size of the wire on the edges of the apron is slightly larger (and in bigger coils) than the size of the wire on the decorative pieces. You can use 0.7- 1.3 mm (16-20 gauge) wire for this.
In short- The later time periods used smaller wire than the earlier time periods. The interior decoration used smaller wire than the border. The interior decoration was a later development than the border. Not everyone would have an interior decoration either.
The wire should be round, from what I can tell.
As a source, I like: https://www.etsy.com/listing/161646629/bronze-wire-premium-half-hard-round . It’s a reddish/ coppery-colored bronze. The seller answers questions and ships quickly. After your first order, she’ll give you a 10% discount code, so you might as well just buy a single roll to make sure you like it, and then order more with the code (shipping is free)
See the comment that is posted below for a better source than what I had.
You will need more than you think, because the coils are pretty dense, and they use up wire quickly.