what is "Minimum Length Least Square" I am in the process of implementing Bayesian Lasso with Normal-Gamma prior; In section 3.3 mention
The prior for the scale parameter $\gamma$ conditional on $\lambda$ is given by $v_\beta = 2 \lambda \gamma^2 \sim IG(2, M)$ where $IG$ denotes the  inverted gamma distribution, the inverse 
of a gamma distribution, so that $IG(2, M)$ has expectation $M$; When $p > n-1 $; $M=\frac{1}{n}\sum_{i=1}^{p} \beta_i^2$ where $\beta_i$ is the Minimum Length Least Squares (MLLS) estimate. 
I am not sure, if "Minimum Length Least Squares" is same as "Least Square Solution"
 A: A least squares problem (linear or nonlinear but I'll use a linear example here) can have a unique least squares solution, or it might have infinitely many least squares solutions.  For example, this frequently occurs in the presence of multicolinearity in linear regression.  
In the case where the least squares problem has infinitely many least squares solutions it is often desirable to select the solution that has the minimum euclidean length (2-norm) among all of the least squares solutions.  
For example, suppose that we want to solve $\min \| X\beta-y \|$, where
$X=[1\;\; 0]$
and 
$y=2$.  
In any least squares solution $\hat{\beta}_{1}=2$, but we can pick any value of $\hat{\beta}_{2}$ that we want.  e.g. $\hat{\beta}_{2}=10$ gives a least squares solution, as does $\hat{\beta}_{2}=0$.  Among all of the least squares solutions, $\hat{\beta}_{2}=0$ gives the minimum length least squares solution.  
The short answer to your question is that "the minimum length least squares solution" is not generally the same thing as "the least squares solution."
