You advance in Patronage to the blacksmith by spending money and using the services at her shop, primarily.
From the blacksmith's wiki page:
Furthermore, by making use of her services you will unlock patronage levels that will result in some perks when using her services.
As noted in PowerPyx's trophy guide, to be able to make 60 requests at the Blacksmith, you will need to spend a grand total of 20 million yen at the blacksmith.
You need to spend a total of 20 million money!
Up until 30 requests every 200,000 you spend gives 1 request point. After that it gradually takes more money per request point.
However, there is a way to increase your Patronage without grinding as much money.
As hinted at in the trophy guide, you can potentially farm Patronage from exhausting dialogue with the blacksmith:
Exhausting all dialogue options from Tome and Muramasa also gives points sometimes. For example, when talking to Muramasa he may ask you a question. Make your dialogue choice and check if you earned any request points from it. This doesn’t always happen. Just try it every now and then. If you only have a few requests left this can be a quick booster without having to farm money. Personally. I got 2 x 2 points by talking to Muramasa repeatedly.
While it doesn't seem like it worked out for the writer of the trophy guide, other users seem to have had more luck with this method, as seen in this thread:
I just noticed this today as well. If you keep spamming "Speak to Tome" it seems to randomly award a blacksmith request. I don't understand it at all. I've gotten like 5 requests in the last 5 minutes doing this. [...] I kept doing it to test how long/often it would work and I got about 35 blacksmith requests in 20-25 minutes or so and got the trophy.
In addition to these Patronage increases, @CyberClaw has noted that when Tome changes clothes, she will ask if you like her new clothes. Responding to this will award 10 Patronage points. This has been mentioned in this thread as well, which also suggests there are other dialogues that award multiple Patronage points.
Yeah, I just got three patronage points through talking with her and telling her that her dreams are achievable.