Dave,
As an interim method, i.e. clunky, you can locate individual Bird sites using the "direction-distance" tool in editing functions for drawing polylines and locating vertices. This is activated by snapping the sketch tool with a polyline FC to your control/reference point then right clicking for the pulldown menu or CTRL- G and entering the direction-distance data. Then you would have to snap a point to the end of the line that is drawn. This will work very precisely, and could be sped up by entering the direction and distance info in fields for the reference point so you can see them in the attribute table then draw all the lines before switching to the point feature class to add points at the end of the lines for your birds.
Alternatively, you can do this with the direction-distance editing tool in a point feature class only, (looks like a circle with a line through it in the 9.3 editing tool pulldown). First snap to you reference point and hit A to enter an azimuth, snap to the same point again and hit D to enter the distance, then hit enter to make a new point.
I think I prefer the polyline method; having to snap back to the same point twice is a bit confusing and drawing the polylines will create a verifyable record of how you located the points.
With the points located you can output the coordinates in new fields in your point feature class using Calculate Geometry.
Another way that runs a bit outside ArcMAP, I guess you could export your GPS reference points to a spreadsheet then add your azimuth and distance and calcuate trigonometrically (add D*sin (az in radians) to the X coordinate and D*cos (az in rad) to the Y for new X-Y positions then return that table to ArcMAP as an ADD X-Y Data function from the Tools menu using the new coordinates. You could do the calculation in ArcMAP but you would have to reinput the X-Y table anyway.
good luck,
Hardolph