T1 Contrast
INCORRECT. This answer refers to changing the echo time (TE). The TE enhances or minimises the differences in signal due to differences in the T2 of tissues, not the T1.

T1-weighted spine image. Note the cerebrospinal fluid is dark, and subcutaneous fat is brighter.
CORRECT. The answer—waiting for magnetisation recovery—refers to changing the repetition time (TR). The TR enhances or minimises the difference in signal due to differences in the T1 of tissues.
Recall that we cannot measure an MRI signal unless the net magnetisation vector is pushed away from alignment with the external magnetic field. It then precesses, generating a signal perpendicular to the external magnetic field. If we rotate the net magnetisation vector by 90°, the entire magnetisation is tipped into the xy-plane. The amplitude of the signal (Mxy) then decays away exponentially with a time constant T2, and the Mz component of the net magnetisation vector regrows with a time constant T1.
Consider a single NMR signal measurement: rotation of the net magnetisation vector into the transverse plane, waiting a time TE, and having a certain signal available to measure. However, in MRI, a large number of such measurements are required (e.g. 256), most of which must be measured before an image can be calculated. When the net magnetisation vector is moved into the transverse plane, it is necessary to wait an amount of time to allow Mz to recover, ready for the next rotation into the transverse plane for the next signal measurement.

T1 recovery curves (Mz(t) = M0(1-exp(-t/T1))).
If we wait a only a short amount of time since the last rotation (e.g. 90° rotation) before measuring the signal (a short TR), then a large difference between the magnetisation of tissues will have developed. However, if a longer time is allowed to pass (a longer TR), the difference between the signal amplitudes of tissues will have have reduced.
Thus, a shorter TR enhances differences in signal arising from differences in the T1 of tissues (even though a longer TR would provide more magnetisation from both tissues to be rotated into the xy-plane for measurement). Using a shorter TR achieves T1-weighting.
The TR is an operator-selectable parameter on the MRI scanner. Contrast differences due to T1 in a MRI image can be manipulated.
INCORRECT. The native T1 relaxation time of a tissue cannot be changed (for a specific magnetic field strength).
Further reading on this topic:
Books: Q&A in MRI p31, MRI From Picture to Proton p152, MRI The Basics p41
Online: Medcyclopaedia, Basics of MRI