First, you need to contact the museum's curator and ask for permission. In correspondence, be polite, honest, brief, and let the curator know that you will accommodate his/her schedule.
So. Now you have permission to document measurements and photos of your beloved artifact. What do you need? How do you prepare? All you need is a ruler, tape measure, notebook, pen, and camera, right?
Yes and no.
That's all that accompanied me on my appointment with the scheitholts, and I was ill prepared....and nervous that I would take up too much of the director's time. So much so in fact, that I forgot to take the measurements of the overall length of both zithers!
This morning I heard back from the curator at the Gettysburg Military Museum, granting me permission to photograph and measure Robert E. Lee's Medicine Box (after December).
I haven't been this excited since the Christmas I unwrapped my "Space 1999" action figure set and model spaceship.
This time, I'll take a worksheet with me that lists all the parts of the cabinet that need to be measured and columns for wood/other material, width, length, height, and thickness. I'll also sketch the cabinet (it helps if you can take a snapshot before your appointme

The other thing I'll do is not get too creative with the photos (like I did with the zithers). I'll be sure to take lateral, top & bottom, and straight on shots of each part of the cabinet. That way, if I forget to take a measurement, I can refer to the photo and use a known measurement to obtain the missing one.
Have I forgotten anything? Probably. But it's an exciting learning experience of galactic proportions.