How to guess the content of a christmas present? Let us assume that the present does not make any recognizable sounds when shaken (meow splat - the present now contains a dead kitten). Let us furthermore assume that the internal state of the present does not change, so that any measurements can be repeated.
I will consider it cheating to unwrap the present, or shine any form of light, rays or particles through the present. Not many people has x-ray or neutron scattering devises available anyway. - and obviously it is cheating to squeeze the present.

What can be measured? 


*

*Size and shape of wrapping(hey mom ... my wish elephant will not fit in here)

*The mass 

*Position of center of mass

*Moment of inertia tensor

*Vibrational resonance 

*Magnetic resonance 

*Thermal radiation



If the seven scalars in mass, position of center of mass and moment of inertia tensor were measured, what could be said about the shape and mass distribution of the present?
If vibrational or magnetic resonance were measured, what could be said about the shape and materials of the present?
Would the thermal radiation not just depend on the temperature of the wrapping? what if the room temperature were dropped, then the present would heat the wrapping depending on the heat capacity, and thermal conductance. -but can it be used for anything?

This question has been heavily edited to correct my previously mistakes, and reflect my progress. THANK YOU for the answers already given.
 A: Rotating about an axis not going through the center of mass does not give anything new. There is Huygens-Steiner theorem saying that moment of inertia with respect to any axis is
$$
I' = I_\text{c.m.} + m d^2
$$
where $I_\text{c.m.}$ is the moment of inertia about the center of mass and $d$ is the distance between new axis and the center of mass (the axes must be parallel).
The general expression for the moment of inertia tensor about a displaced axis can be found here.
Hence the measurements of the moment of inertia tensor about multiple axes will give only the tensor itself and position of the center of mass. A solid brick of appropriate shape placed in the center of mass will give the same result. 
A: Everything you can do (other than “shaking” — mechanical interactions) is going to involve “light, rays or particles”, because one of the principles of physics is locality; any information you get about the inside of the package must be from something that propagated out of it. Nearly anything you can do is going to involve “shining light”, that is, photons (= electromagnetic waves), though they may be extremely low frequency.


*

*You can in principle measure the gravitational field produced by the contents of the present, and thereby obtain more information about the mass distribution. However, this is an extremely weak effect and entirely impractical; its only advantage is that it definitely does not involve any photons (but it would involve gravitons, if they exist).

*Change the ambient temperature and observe the pattern of thermal radiation emitted by the package, using a thermal camera. This is “shining light” but only light that is already going to be there.

*Place the package in an electric or magnetic field and observe the forces produced on it and how it alters the field. The refined form of this is NMR/MRI imaging; however, that involves some high-frequency electromagnetic waves. If you want to stick to low frequencies and everyday equipment, you could move a magnet around the package and note any attraction. (Or rather, for sensitivity, hang the magnet on a string and move the package next to it.)

*Shake it better: aim a speaker at it, and examine the frequency analysis of the response (compared to the sound without the package present). This can tell you the resonant frequency of shapes inside the present and therefore some guess as to, e.g., the size of flat surfaces. The refined form of this is ultrasound imaging.
A: The measurements which you have allowed only support the determination of a very limited number of variables.
Note that the moment of inertia w.r.t. axis parallel to vector n=[n1, n2, n3] equals
\begin{equation}
I_n= n^{T}In=\sum_{i=1}^3 \sum_{j=1}^3 I_{ij} n_i n_j
\end{equation}
where In is the moment of inertia w.r.t. axis n, I is the inertia tensor and Iij is its corresponding matrix element.
Thus, no matter how many measurements of moment of inertia w.r.t. to different axes one performs, all these measurements yield just the inertia tensor, which is merely 9 numbers. Also, it can be shown that I can be represented by a symmetric positive semi-definite matrix. Spectral theorem for symmetric positive semi-definite matrices ensures the existent of a basis in which the matrix is diagonal:
\begin{equation}
I = \left[
\begin{array}{ccc}
I_x & 0 & 0 \newline
0 & I_y & 0 \newline
0 & 0 & I_z
\end{array}
\right]
\end{equation}
where Iv represents the moment of inertia w.r.t. axis v and x, y, z is the basis composed of eigenvectors of I (in the context of inertia tensor x, y, z are also called principle axes of I).
This means that all the measurements of moments of inertia w.r.t. different axes yield information that can be represented with just 3 (real non-negative) numbers. We disregard the information about the orientation of the principle axes here assuming that presents that can be transformed into one another via rotations can be identified. As for presents which are each others' mirror reflections, they cannot be distinguished by measuring moment of inertia.
The measurement of center of mass yields additional 3 numbers: the coordinates of the center of mass. The measurement of the mass, obviously yields just one more number. Thus all the allowed measurements yield a total of 7 numbers.
It is not difficult to imagine presents whose structure cannot be encoded on 7 numbers alone. All the presents most children would want have this property. Thus, the allowed measurements cannot tell many of these presents apart.
Measurements of moment of inertia with respect to parallel axes don't give much more information due to parellel axis theorem.
A: What about using an ultrasound machine?

Ultrasonic Testing (UT) uses high frequency sound energy to conduct examinations and make measurements. Ultrasonic inspection can be used for flaw detection/evaluation, dimensional measurements, material characterization, and more. 

