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As suggested by Budney and Agol's answers, one short answer to the question "why categorify [3-manifold invariants]" is "to get 4-manifold invariants."

This is directly related to the standard answer to the question "why is homology better than the Euler characteristic" which is: "functoriality." Functoriality in the case of Khovanov homology leads to maps coming from 4-dimensional cobordisms between knots, which one hopes will give interesting information about knot cobordisms (this has been realized in the work of Jake Rasmussen).

More specifically, the work of Donaldson/Floer/Witten et al in the 80's and early 90's led to an invariant for 3 and 4 manifolds which you might call "Donaldson-Floer theory" and which (more or less) fit together into what came to be called a "3+1 dimensional topological quantum field theory." D-F theory gave many exciting results about the smooth structure of 4-manifolds, but it was defined using complicated non-linear analysis and was very hard to compute.

Around the same time, there appeared another TQFT, the Witten-Reshetikhin-Turaev (WRT) theory. This was a 2+1 dimensional TQFT, and the work of Reshetikhin-Turaev showed that it had a purely combinatorial definition. Moreover, it gave invariants of knots which were directly related to the Jones polynomial.

According to the introduction of Khovanov's paper, his original motivation goes back to ideas of Crane and Frenkel (CF) on lifting the WRT theory from a 2+1 to a 3+1 theory, in the hopes of getting a combinatorially-defined invariant of smooth 4-manifolds with the same power as D-F theory. To understand, from their point of view, why this is related to "categorification" in the algebraic sense (a term which they may have helped define? I'm not sure), see their paper "Four dimensional topological quantum field theory, Hopf categories, and the canonical bases." There, they say

"the second idea motivating our construction is that replacing an algebraic structure with a similar categorical algebraic structure lifts the dimension of the corresponding TQFT by 1"

Currently, Khovanov homology is not defined for all knots and knot cobordisms in all 3 and 4 manifolds, and it gives limited (albeit very interesting!) 4 dimensional information. It remains a fundamental problem to get a complete combinatorial 4-manifold invariant generalizing Khovanov homology and fit it into a TQFT, and to understand its relationship with (categorified) representation theory and smooth 4-manifold topology.

I should add that there is a different point of view on this question, which takes the "categorification of quantum groups" as a fundamental goal and interesting problem in its own right. But this should probably be addressed by someone who knows more about it than I do...do!
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As suggested by Budney and Agol's answers, one short answer to the question "why categorify [3-manifold invariants]" is "to get 4-manifold invariants."

This is directly related to the standard answer to the question "why is homology better than the Euler characteristic" which is: "functoriality." Functoriality in the case of Khovanov homology leads to maps coming from 4-dimensional cobordisms between knots, which one hopes will give interesting information about knot cobordisms (this has been realized in the work of Jake Rasmussen).

More specifically, the work of Donaldson/Floer/Witten et al in the 80's and early 90's led to an invariant for 3 and 4 manifolds which you might call "Donaldson-Floer theory" and which (more or less) fit together into what came to be called a "3+1 dimensional topological quantum field theory." D-F theory gave many exciting results about the smooth structure of 4-manifolds, but it was defined using complicated non-linear analysis and was very hard to compute.

Around the same time, there appeared another TQFT, the Witten-Reshetikhin-Turaev (WRT) theory. This was a 2+1 dimensional TQFT, and the work of Reshetikhin-Turaev showed that it had a purely combinatorial definition. Moreover, it gave invariants of knots which were directly related to the Jones polynomial.

According to the introduction of Khovanov's paper, his original motivation goes back to ideas of Crane and Frenkel (CF) on lifting the WRT theory from a 2+1 to a 3+1 theory, in the hopes of getting a combinatorially-defined invariant of smooth 4-manifolds with the same power as D-F theory. To understand, from their point of view, why this is related to "categorification" in the algebraic sense (a term which they may have helped define? I'm not sure), see their paper "Four dimensional topological quantum field theory, Hopf categories, and the canonical bases." There, they say

"the second idea motivating our construction is that replacing an algebraic structure with a similar categorical algebraic structure lifts the dimension of the corresponding TQFT by 1"

Currently, Khovanov homology is not defined for all knots and knot cobordisms in all 3-manifolds3 and 4 manifolds, and it gives limited (albeit very interesting!) four 4 dimensional information. It remains a fundamental problem to get a complete combinatorial 4-manifold invariant generalizing Khovanov homology and fit it into a TQFT, and to understand its relationship with (categorified) representation theory and smooth four-manifold 4-manifold topology.

I should add that there is a different point of view on this question, which takes the "categorification of quantum groups" as a fundamental goal and interesting problem in its own right. But this should probably be addressed by someone else..who knows more about it than I do...
show/hide this revision's text 1

As suggested by Budney and Agol's answers, one short answer to the question "why categorify [3-manifold invariants]" is "to get 4-manifold invariants."

This is directly related to the standard answer to the question "why is homology better than the Euler characteristic" which is: "functoriality." Functoriality in the case of Khovanov homology leads to maps coming from 4-dimensional cobordisms between knots, which one hopes will give interesting information about knot cobordisms (this has been realized in the work of Jake Rasmussen).

More specifically, the work of Donaldson/Floer/Witten et al in the 80's and early 90's led to an invariant for 3 and 4 manifolds which you might call "Donaldson-Floer theory" and which (more or less) fit together into what came to be called a "3+1 dimensional topological quantum field theory." D-F theory gave many exciting results about the smooth structure of 4-manifolds, but it was defined using complicated non-linear analysis and was very hard to compute.

Around the same time, there appeared another TQFT, the Witten-Reshetikhin-Turaev (WRT) theory. This was a 2+1 dimensional TQFT, and the work of Reshetikhin-Turaev showed that it had a purely combinatorial definition. Moreover, it gave invariants of knots which were directly related to the Jones polynomial.

According to the introduction of Khovanov's paper, his original motivation goes back to ideas of Crane and Frenkel (CF) on lifting the WRT theory from a 2+1 to a 3+1 theory, in the hopes of getting a combinatorially-defined invariant of smooth 4-manifolds with the same power as D-F theory. To understand, from their point of view, why this is related to "categorification" in the algebraic sense (a term which they may have helped define? I'm not sure), see their paper "Four dimensional topological quantum field theory, Hopf categories, and the canonical bases." There, they say

"the second idea motivating our construction is that replacing an algebraic structure with a similar categorical algebraic structure lifts the dimension of the corresponding TQFT by 1"

Currently, Khovanov homology is not defined for all knots in all 3-manifolds, and it gives limited (albeit very interesting!) four dimensional information. It remains a fundamental problem to get a complete combinatorial 4-manifold invariant generalizing Khovanov homology and fit it into a TQFT, and to understand its relationship with (categorified) representation theory and smooth four-manifold topology.

I should add that there is a different point of view on this question, which takes the "categorification of quantum groups" as a fundamental goal and interesting problem in its own right. But this should probably be addressed by someone else...