Interpretation of lagrange multiplier
As mentioned in the other answer, the Lagrange multiplier is the marginal effect on the value (optimized) function, when the constrained is "relaxed" marginally. In your case then it should be interpreted "how much studying time changes as the required average expected grade..." increases?

Well, this is a good example to show that how we set up the Lagrangian function matters in order to interpret what we get meaningfully. Your is a minimization case so we have 
$$\min T = t_1+ t_2 \\
s.t\;\;\; \displaystyle m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}=0$$

How do we write the Lagrangian? Do we write

$$\Lambda = t_1+ t_2 + \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

or

$$\Lambda = t_1+ t_2 - \lambda\left[m-\frac{[{g}_{1}({t}_{1})+{g}_{2}({t}_{2})]}{2}\right]$$

Since the constraint is an equality constraint, you may hear that "it doesn't matter". Mathematically, it indeed doesn't, but it does matter when the time comes to interpret the value of the multiplier. And, maybe in Utility maximization problems we don't much care because utility is ordinal. But in a problem like the one the OP is solving, the objective function is measured in a very real and measurable and quantifiable unit, time.

Now how we should understand "what happens when the constraint is relaxed"? In the specific minimization problem, it intuitively means "what happens when the required $m$ is reduced". (Since "relax" has a "desirable" connotation, and so we would desire to be asked for less effort, a lower target). 

I guess it is now clear how the obtained value for $\lambda$ can be interpreted. And do ponder what variant of the Lagrangian is the appropriate one for your case.