Thanks to Vince's comments I was able to perform succesfully the whole process. 

As I wrote, first I wanted to establish a number of AOIs on the raster. Nothing fancy here, I got raster properties by using this small function below:

    public static IRasterProps GetRasterProperties(IRasterDataset rasterDataset, int rasterBandIndex)
    {
         IRasterBandCollection rasterBands = (IRasterBandCollection)rasterDataset;
         var rasterBand = rasterBands.Item(rasterBandIndex);
         return (IRasterProps)rasterBand;
    }

Given the raster properties, I created a number of areas each defined by an IEnvelope. In order to split the input raster to AOIs, I used the ExtractByRectangle Geoprocessing tool. 

    ExtractByRectangle extract = new ExtractByRectangle(inputRaster, envelope, path);
    IGeoProcessorResult2 result = gp.Execute(extract, null) as IGeoProcessorResult2;
    
    parameters:
    inputRaster - IRasterDataset (input raster)
    envelope - IEnvelope (definition of our AOI)
    path - file path to resulting raster

I won't describe all Geoprocessing shenanigans, but that is where ArcObjects get very cryptic in my opinion. A number of error codes that have no obvious explanation doesn't help too. I found great code by Kirk Kuykendall that helps a lot with debugging here: http://gis.stackexchange.com/questions/1610/avoiding-fails-from-arcobjects-geoprocessing-with-net

Now, we have our small raster AOIs. I need to get the point of maximal elevation for each of them. Hence, I start with computing statistics for every AOI using the function below:

    public void ComputeRasterStatistics(IRasterDataset rasterDataset, int rasterBandIndex)
    {
         IRasterBandCollection rasterBands = (IRasterBandCollection)rasterDataset;
         var rasterBand = rasterBands.Item(rasterBandIndex);
         RasterStatistics = rasterBand.Statistics;
    }

Statistics deliver information concerning values of raster extremes, inclusing the maximum point value of a raster. Next I convert the raster to points using the function below.

    public static IFeatureClass RasterToPoints(IRasterDataset raster)
    {
         IConversionOp conversionOp = new RasterConversionOpClass();
         IWorkspace shapeWS = FeatureWorkspaceHelper.CreateInMemoryWorkspace();
         var featClass = conversionOp.RasterDataToPointFeatureData((IGeoDataset)raster, shapeWS, Guid.NewGuid().ToString());
         return (IFeatureClass)featClass;
    }

FeatureWorkspaceHelper.CreateInMemoryWorkspace(); is a helper function of mine, that creates an empty in-memory workspace. Have in mind this can be obviously done with 'classic' file workspace too.

Now I just need to find a point that has the maximal elevation and return it. The function below (some hardcodes there!) does that.

    private IPoint GetRasterMaxElevationPoint(IFeatureClass featureClass, double val, int elevationIndex)
    {
    	IQueryFilter queryFilter=new QueryFilterClass();
    	queryFilter.WhereClause = "GRID_CODE >= " + (val - 0.01).ToString();
    
    	var cursor = featureClass.Search(queryFilter, true);
    	IFeature feature = null;
    	IGeometry shape = null;
    	double maxValue = double.MinValue;
    	while ((feature = cursor.NextFeature()) != null)
    	{
    		if ((double) feature.Value[elevationIndex] > maxValue)
    		{
    			shape = feature.Shape;
    			maxValue = (double) feature.Value[elevationIndex];
    		}
    	}
    
    	return new PointClass()
    	{
    		X = shape.Envelope.LowerLeft.X,
    		Y = shape.Envelope.LowerLeft.Y,
    		SpatialReference = shape.SpatialReference
    	};
    }