How do you come up with ideas for graphics when designing a website?

Find some inspiration. Pick a style you like and do a little research on it. Or, if working on a brand, start with the brand guidelines, there is usually good content in there.

The next step for me is the content. Write the content for the web page in plain text. Like you are reading it top to bottom. That will show you the structure of your web design. Then you can see what phrases/statements are begging for some supporting graphics.

Sometimes a design starts by just using a shape from the brand in a creative way. If the logo has a rounded rectangle in it, try making a 45 deg version of it that is big and faded in the top right corner of the page. Now put your opening headline over it on the left.

Your website should have a theme and Topic, Follow that, what graphics go with your theme and topic, if it's a personal website it should be about you or the person it covers, if it's geographic cover photos, activates that geographic, if a product website, add graphics/animation appropriate to the product. Deciding those should be the first thing.