grammar - "More than (what) meets the eye"

Both options are perfectly acceptable. One is a noun-phrase (NP) complement of than and the other is a comparative-clause complement. They are structurally equivalent to

1. there is more than a budget cut


2. there is more than was included here.

In [1], a budget cut is an NP complement of than. Similarly, what meets the eye is also an NP complement of than.

(That's all you need to know about that construction, but if you want to unpack it a bit more, what meets the eye is like the thing that meets the eye. This is an NP with thing as its head and a relative clause functioning as a modifier. In the case of what meets the eye, the head and the relative are fused: the pronoun what functions both as the head of the NP and the subject in a modifying relative clause.)

The other option is there is more than meets the eye. In this case, meets the eye is a standard comparative clause functioning as complement of than. A comparative clause is different from a main clause in that it typically has a gap (e.g., I'm bigger than [she is _], in which the adjective big is gapped out.) The gap in your sentence is in the subject position (i.e., there is more than [_ meets the eye]) and could be filled by the NP the budget cut, for example, if it were a main clause.