The individual from Thyatira that most Christians know about is Lydia, who came to faith in Christ when she heard Paul in Philippi (Acts 16:14). It is not known whether she was involved in helping commence the church in Thyatira, although we can assume that she would have identified with it when she was in the city.
Apparently, Thyatira was the least important of the seven cities, which then makes the fact that it received the longest letter something to think about. Maybe Jesus was saying to the church that while the city was not of much consequence in the eyes of the emperor such could never be said of a church and its Lord. Churches in unimportant places are important to Jesus.
The description of Jesus that is given to Thyatira reminds them that he is a divine speaker, with penetrating knowledge and purity and strength. The implication is that his words will come with precision and with power. Jesus is contrasting his speech with the teaching given by the false prophetess who had influence in the church there.
There are usually connections between details in each of the seven letters and what went on in the relevant city. So we can ask why Jesus referred to himself as the Son of God. An answer could be that a focus of the pagan worship in the city was on Apollo, who was regarded as a son of Zeus. Perhaps those criticised in the letter went to activities connected to the false god. If so, they were reminded that whereas Apollo could not see anything Jesus could see everything, a reminder that there is a huge gap between false religion and the religion of Jesus.
One of the trades that was prominent in Thyatira was bronze-work and there may be an allusion to that trade in the description that Jesus makes of his feet. He does not say that his feet are burnished bronze, but like it. The meaning of having feet like this could be a realisation that Jesus can treat his opponents with great power.
It is also likely that the feet of Jesus are being compared to other feet that walked in the church. In contrast to the feet of the false teacher who led her followers into sinful practices, the path that Jesus walks is one of purity because he is pure. Saying his feet are like bronze could be an allusion to the way that metals were purified. Of course, Jesus did not need to be purified because he was always pure. Because he is pure, he will speak pure words about what he sees in the church.