The cost effectiveness of the dollar would depend a lot on how much of a renovation is needed or desired.
I would recommend seeking out a general contractor to come to the property and give you an estimate based on what you are wanting and what the home is needing renovation wise this will tell you your renovation cost. As far as the cost of new construction, you can typically look at new construction sales in your area and see what the cost per square foot is. Take that number and multiply it by the square footage you are seeking and that will give you an idea of what your new construction costs will be.
I won’t get to deep into the cost comparisons. With very few exceptions it will always be less total dollars spent on a renovation than a tear-down and rebuild. You should research the resale comps in your area to evaluate the ROI.
My first advice would be to research building and zoning requirements. In our area, most notably waterfront properties and urban in-fill, it is not uncommon to find newly implemented restrictions on property use. New builds are subject to all new zoning restrictions. With renovations an owner is typically allowed to conform to original land use codes (grandfathered).
Best of luck.