There's no magic way to estimate the cost of a project, but if you use two or three ways and then compare them you are probably more likely to get nearer to reality.
From the ground up, activity by activity.
Make a list of every activity that forms part of the project (and you'll probably miss a few so maybe you should add 10% to the
finished total) and costs for those. Smaller things are usually easier to estimate. If stuck on an activity, break it into
Maybe there is an expert you can ask, or get them to check your estimates.
Maybe a similar project has been done before, or at least one with some of the tasks being similar, so you can compare your estimates.
Team size x Time.
How many people are you planning to use, and how long do you think the whole thing will take, roughly? Then multiply the two. If the numbers are going to vary (often projects start slowly and then rise to a peak with lots of people, and then the numbers of people dwindle in the finishing stages again) then take an average.
This method seems pretty rough but if you are experienced it can be a good double check for method 1.
Have a guess!
I know this sounds rather unscientific, but from experience you'll know roughly the size of the project compared to others. Someone will have probably set an approximate budget based on this - after all, the question 'How much money is this project worth to us?' should always be asked, probably before any detail pricing.
It might be interesting to do this first, and then see if Method 1 comes in similar.
-> if all three are not the same you've got a problem. And the problem might be with method 1!.