I was sent here by u/iphikrates for your Early Iron Age Greece needs.

The answer to this question is a little bit complicated, because it could be loads of evidence, or it could be just a little bit, depending on how you interpret a certain class of object.

The objects in question are usually called phalara (sing. phalaron) and they usually look like a raised dome with a circular edge. They're typically bronze (I know of one iron example, from Athens) and they first appear in the archaeological record in the 12th century in graves with weapons. They are usually interpreted as shield bosses or omphaloi because of their shape. But the shapes aren't exactly consistent. Sometimes they are simple domes with an edge; other times they have a long spike-like appendage sticking out of the middle.

The Homeric passages you mention are pretty much the reason why they are interpreted as shield bosses. However, in Iron age graves from Macedonia very similar, but smaller objects are found in the graves of women. These have been interpreted as belt attachments. Another example from Albania, which includes small and large phalara, has been reconstructed as body armour. Another, from Cyprus, found with various strips of bronze, has been reconstructed as either a shield OR body armour.

By the eighth/seventh century, they appear in some numbers as dedications at Olympia. These were originally understood as cymbals, although they are never discovered as pairs. They might, however, be musical instruments that were dangled from a hoop of perishable material and struck with a stick, as they usually have holes at the top of the dome.

In Iron Age Athens there are three examples from graves, used as lids for the cremarion urn. None of them were found with other weapons. As such, I tend to think of these as lids rather than shield bosses. They may have come from shields, but I am not convinced that is relevant (the size and shape of the graves and the probable weight of the shields makes it unlikely that they were still attached to the main body when buried).

In the 1970s archaeologist Anthony Snodgrass suggested that we cannot be certain that these are a single class of object, and I tend to agree with him on this one. Some of these phalara, such as those found with weapons, were probably shield bosses. Others were likely belt or armour attachments. Some may have been musical instruments. Other interpretations include horse armour (I find that dubious because of lack of evidence, personally) and my own suggestion that some are literally just lids.

As for iconography, it is more limited. These bosses wouldn't have been used on the Dipylon shields, and I can't think of any clear examples from Late Geometric vase painting. This may mean that the boss-style shields were being replaced by other types at that time.