Having a budget in mind is great, but when the person buying something with said budget does't know how much they need to spend to get what they want, and they don't know what they actually want, then having a budget is also a bit pointless.
Start at the beginning. What is the computer going to be doing? Only gaming? If so what type of games will they be playing, and how long do they expect the system to last before investing further in it?
Next, chose the monitor resolution and refresh rate that you'd like to achieve, then look and ask is that realistic. Someone always say 4k 144Hz HDR RGB-Sync Dolby Vision ZOMG WTFBBQPWN. When they'd be very happy with 3440x1440p at 120Hz Freesync.
Then finally, look at what budget you have left after finding the right monitor, and work out what will give you the best £:FPS spent over the chosen XX period before altering any components.