Element of promoter responsible for expression power of gene
Tldr; In an in vivo context the answer is going to be dependent on DNA sequence and context.

Simplistically, Eukaryotic genes promoters can be segmented in to several parts. The first, a core promoter region where RNA pol II will bind. This isn't capable of a robust transcriptional response on its own and relies on proximal promoter regions, or more specifically transcription factor binding to the proximal promoters. Proximal promoters are usually 100 or so base pairs upstream of the transcription start site and will bind proteins that will assist in Pol II binding and activation (or silencing). Enhancers (and Silencers) are conceptually similar to proximal promoter regions with the exception that they can be a looooooooooong way away from the transcription start site in sequence space. See here:1.

So, if you just wanted to drive expression of a transgene in a cell line in the lab there are plenty of established cloned promoters and enhancer regions known to work such as SV40 or Gal4. This would be an example of "replacing the promoter". That's exactly what it is and why these sequences were chosen to be cloned.

If you are asking which regions are specific to activate or repress a gene in vivo and/or in context then it will be, of course, context dependent and you'll have to find out yourself if someone hasn't already. Generally, these regions of effect will be binding sites for sequence specific transcription factors  which interact with the core transcriptional machinery (or bind other proteins that do) to effect the rate of transcription. So, if the sequence is what can induce an effect, how can you find it? You can search for these sequence specific sites by analyzing your sequence of interest with a slew of computational tools for potential binding sites and directly assess any hits ability to increase/decrease expression as isolated units when cloned upstream of an appropriate reporter construct. I guess you could even go as far to CRISPR single base pairs in or out to "edit" any hits. Alternately, you could use a purely wet bench technique such as promoter bashing to similar effect - performing serial deletions to find out which part of your promoter is important for strong/medium/weak effects or activation/repression. 

The Stark Lab developed a cool genome-wide approach looking at the ability of DNA to drive expression. Although, something worth bearing in mind is that in a normal in vivo situation everything is context dependent - one piece of sequence may be reliant on another close by (like the core promoter requires proximal promoters) or it may require one or more pieces of DNA that are distant in sequence space to be close by in 3D space (like distal enhancers).

People have also taken completely synthetic approaches to making promoters .

This would be easier to answer if your question was more specific. 
HTH.