I see your point... The problem is that normal human tissues are very hard to get, you are not doing biopsies for no reason (they are illegal!)... You usually extract sick tissue: cancer, atherosclerotic plaques, benign tumors, etc. 

I still think you need to start with cell cultures, animals models and look in epidemiological studies. One option is to study autopsies from people who die via trauma to obtain normal tissue. Otherwise, you'll results be biased..

You also need a very strong hypothesis (unless you were try to fish something)...

Two sites with good tissue-specific information for gene expression:

http://www.proteinatlas.org/ (RNA-Seq and immunostaining)

http://ist.medisapiens.com/ (microarray)

Besides the Protein Atlas already suggested above (this one is good because it's provides also protein level information and cellular localization), you can take a look also to https://www.proteomicsdb.org/, that also have expression information for both mRNA and proteins.

If you are interested only in gene level expression I suggest the excellent EBI Expression Atlas. THis use RNA-Seq information from different projects that have mapped gene expression across several human tissues, organs and cell types (like the Illumina Body Map). You can search for gene but also for specific tissues and get the list of genes expressed in the relevant district above a certain threshold. You can also search for gene that are exrpessed specifically in a given tissue (based on threshold between the expression in the tissue of interested versus other samples).

Finally the recently published GTeX data provides the most complete and updated information on gene expression across human tissues.  Moreover it also provides information on eQTLs, http://www.gtexportal.org/home

However you should note that, besides some special tissue like brain and testis, most of the genes are found at different expression levels across many (if not all) tissues. So you'll have to set a threshold to select those genes that you consider interesting for your study.

Good luck.

Edoardo, thanks for sharing those links.  The GTeX dataset is new to me and looks like a great resource.

The FANTOM5 consortium have (by far the most complete) set of primary tissue and primary cell covering the whole human body.  All these cells/tissues were profiled  using the high-throughput sequencing method CAGE, which identify transcription initiation and thereby also captures gene expression (actually it also enables identification of enhancers and their expression levels, which you might want to consider - take a look at the SNP analysis in the enhancer paper).

Take a look at:

http://fantom.gsc.riken.jp/5/ # for data - the data you wan tis in phase 1

http://www.nature.com/nature/journal/v507/n7493/full/nature13182.html # promoter article

http://www.nature.com/nature/journal/v507/n7493/full/nature12787.html # enhancer article

And a cool tool for extracting tissue specific promoters/enhancers:

http://pressto.binf.ku.dk/

Hope this helps!