Why is gadolinium specifically used in MRI contrast agents?
The reasons may possibly be:


T1shortening
Relatively less adverse reactions
Increased contrast enhancement



  The most commonly used clinically approved contrast agents for MR
  imaging are gadolinium-based compounds that produce T1 shortening.


Tissue relaxation results from interactions between the unpaired electron of gadolinium and tissue hydrogen protons, which significantly decrease the T1 of the blood relative to the surrounding tissues.

Additionally, adverse reactions to this agent are far less frequent than those seen with iodinated compounds, with common reactions including nausea, vomiting, headache, paresthesias, or dizziness.


  T1weighted sequences are part of almost all MRI protocols and are
  best thought of as the most 'anatomical' of images, resulting in
  images that most closely approximate the appearances of tissues
  macroscopically, although even this is a gross simplification.




Figure 1-10. MR enterography on a patient with Crohn disease. A. Coronal T2-weighted image shows wall thickening
and stenotic ileum (arrow). B. Coronal gadolinium-enhanced fat suppressed T1-weighted image shows increased
contrast enhancement of one thickened segment of ileum (arrow).

Credits Basic Radiology Chapter 1 page 11

Update

Most of the answer above was based on Pharmacological/Medical aspects but some Chemistry aspects have been included below:

Also the article mentioned by@Linear Christmas contains valuable information on the subject

Contrast agents


  Contrast agents can be divided into two groups depending on whether
  they cause changes in either T1 (longitudinal relaxation – in simple
  terms, the time taken for the protons to realign with the external
  magnetic field) or T2 (transverse relaxation – in simple terms, the
  time taken for the protons to exchange energy with other nuclei)
  relaxation rates of the water protons, these being known as positive
  or negative agents respectively.
  
  The ability of an agent to affect T1 and T2is characterised by the
  concentration-normalised relaxivities r1 and r2respectively. These
  parameters refer to the amount of increase in 1/T1 and 1/T2
  respectively, per millimole of agent, and are normally quoted as a
  rate (mM21 s21).4 The values are used to determine the efficiency of a
  contrast agent, and they consist of contributions from both inner
  sphere and outer sphere relaxation mechanisms Negative contrast agents
  influence the signal intensity by shortening transverse relaxation
  (T2), thereby producing darker images as high T2 results in increased
  brightness of the images.


Gadolinium (III) vs other Lathanide (III) ions


  Gadolinium(III) reagents are commonly focused on, due to the coupling
  of a large magnetic moment with a long electron spin relaxation time
  of $\ce10^{-9}$s  at the magnetic field strengths used in MRI
  techniques.


Gadolinium (III) ion is unique in a number of respects:

It has the ability to shorten both the longitudinal and transverse relaxation times of water protons approximately to the same extent by relaxing all nearby protons. Other Ln(III) ions with large magnetic moments are less efficient in shortening T1.


  The Gadolinium ion is also unique among other Ln(III) because its
  symmetric seven electron ground state results in an electronic
  relaxation rate that is six orders of magnitude slower than the
  other Ln(III) ions. This in turn results in a strong magnetic
  anisotropy and fast electronic relaxation states with very short T1,
  on the order of $\ce10^{-13}$s.


Other Ln (III) ions tend to undergo Curie relaxation enhancement arising from the interaction of the nuclear spin with the thermal average of the electron spin. (The Curie-spin relxation effect is significant at lower temperatures and higher magnetic fields and for ions with a large magnetic moment).

It has been observed that the Curie spin relaxation effect affects the transverse relaxation more than the longitudinal relaxation. Consequently paramagnetic Ln(III) other than Gd (III) are less efficient T1 relaxation agents.

Use of other Lanthanide (III) ions as MRI contrast agents

Dysprosium(III) is another lanthanide ion that has been used in
MRI, being classed as a negative contrast agent. In the use of high magnetic fields, 


  Gd(III) based contrast agents exhibit poor water relaxivity. As a
  result, the interest in dysprosium-based complexes is increasing as
  they display slow water exchange, due to the need to lengthen the
  residence time in order to optimise the r2 relaxivity. Dy(III) has a
  large magnetic susceptibility which induces local field gradients
  resulting in a lowering of T2.


Generally, dysprosium complexes where the water molecules have a long residence
time, possess potential application as negative contrast agents
at high magnetic fields due to their efficient transverse
relaxivity. However it has been demonstrated that lengthening the residence time of water
can actually be detrimental, because the transverse relaxivity
can then be limiting.

The overall conclusion from the use of dysprosium(III)
complexes as contrast agents is that due to the balancing of
factors required to optimise r2, design of a suitable molecular
structure is crucial. Fine tuning these residence times of the
water protons, and hence the relaxivity, may lead to promising
contrast agents for high field magnetic resonance imaging.


  Europium(II) analogues have been proposed as alternatives to Gd(III)
  because they are isoelectronic—each having seven unpaired electrons.


The only problem with this
complex was that the stability of the Eu(II) chelate was
107 times higher than that of the Eu(III) chelate, which
meant that the Eu(III) complex could be susceptible to
dissociation, possibly releasing toxic Eu(III) into the body.

Other strongly paramagnetic ions e.g Tb(III), Ho(III), Er(III) can have significant effect on line widths and such line broadening usually makes it difficult or even impossible to detect nuclei that are located within a certain distance of the lanthanide ion.

There is still promising use of other Lanthanide (III) compounds as contrast agents as alternatives to using T1 shortening contrast agents as contrast can originate from altering proton density or the total water signal that is detected (using another technique called Magnetisation Transfer (MT).In this case Gadolinium(III)cannot be used in this technique because the T1 of water would be too short, but other lanthanides, such as Eu(III), can be used because they have smaller magnetic moments, relaxing bulk water much less efficiently.

Other potential candidates are still being tested under laboratory conditions, Contrast agents used in the
body clearly should be biocompatible, but there are also other requisites need to be addressed as well. These include the requirements of rapid renal excretion, water solubility, stability in aqueous conditions, and a low osmotic potential when in solution for clinical work.

References


Basic Radiology Michael Y. M. Chen, MD
'Alternatives to Gadolinium-Based Metal Chelates for Magnetic Resonance Imaging' (Chem. Rev., 20102010, 110 (5), pp 2960–3018, 
Lanthanides in magnetic resonance imaging. Melanie Bottrill, Lilian Kwok and Nicholas J. Long
Chemistry Society Reviews, DOI: 10.1039/b516376p