The question I want to ask is vague in a sense. We have examples of mapping class relations, e.g. lantern relation, chain relations, etc. For instance the latern relation on a disk with three boundary components $a$, $b$ and $c$ is: $t_a t_b t_c t_d=t_{ab}t_{bc}t_{ca}$, where $d$ is the curve enclosing all three boundary components and $t_{ij}$ encloses $i$ and $j$ in the most obvious way. While I know how this relation is proved, I have no idea how one comes up with this kind of relations.I want to know if there is general principle to obtain these relations, and whether it is possible to figure out different factorizations of a given element in a mapping class group (of a surface with or without boundary). Is there a more intuitive way to explain for instance the lantern relation?

I can answer your last question: there exist natural ways of embedding the fundamental group of the unit tangent bundle of a surface into the mapping class group, and the lantern relation is the image of an "obvious" relation in the unit tangent bundle group (coming from a visually obvious relation in the fundamental group of a surface). This way of coming up with the lantern relation was discovered independently by myself and by Margalit--McCammond; see our papers

A. Putman An infinite presentation of the Torelli group Geom. Funct. Anal. 19 (2009), no. 2, 591-643.

and

Margalit, Dan; McCammond, Jon Geometric presentations for the pure braid group. J. Knot Theory Ramifications 18 (2009), no. 1, 1–20.

In my paper, this can be found in Section 3.1.2. I also discuss this in my survey paper

A. Putman The Torelli group and congruence subgroups of the mapping class group in "Moduli spaces of Riemann surfaces (Park City, UT, 2011)", 167-194, IAS/Park City Math. Ser., 20 Amer. Math. Soc., Providence, RI.

This occurs towards the beginning (on page 6 in the version on my webpage). All of my papers can be found here.