What type of word is "maybe" that allows it to be placed at the start of a sentence?

Solution 1:

The OED gives maybe as a adverb, noun and adjective. It is most likely to occur at the beginning of a sentence as an adverb.

The etymology is given as

Etymology: Shortened < it may be < it pron. + may v.1 + be v. and can also occur with a subordinate clause, and as a noun.

Maybe was uncommon in English before the mid-19th century when it was considered poetic or dialect. Its origins can be seen in the hyphenated form.

1825 J. Jennings "Observations on the Dialects of West England". May-be, mâ-be, perhaps; for which one of these words is almost invariably used.

1829 J. Hunter "Hallamshire Glossary". May-be. This is at least as good as the hybrid word perhaps, by which it has been supplanted.

As an adverb at the beginning of a sentence - "Maybe I will visit her tomorrow" - it is almost always a free modifier, i.e. it modifies the whole of [what appears to be] the main clause.

As far as

Probably I will. Is not-so allowed.

is concerned. I cannot put it better than Jack O'Flaherty above:

(I don't agree that Probably I will. is impossible, though it's less likely than I probably will.)

Solution 2:

From Merriam-Webster:

sentence adverb (grammar)
an adverb that limits or describes the meaning of an entire statement rather than just a single word or phrase
"Similarly" and "hopefully" often function as sentence adverbs.

I think it's often useful to refer to the usage as whole sentence adverb to emphasize that aspect. Because the adverb applies to the entire containing sentence, it's often just a stylistic choice whether to put it before or after the utterance it modifies. It could also occur within the utterance, but that would often leave more scope for alternative syntactic breakdown / terminology.

I just searched Google Books for the sequence "functions as a sentence adverb" to find some more "typical" examples. Specific examples from that search include sadly, clearly, happily, importantly, but obviously OP's maybe, probably, possibly are all of the same ilk in this respect.

EDIT: I see no real significance to the fact that we're more likely to say...

Maybe that's a good idea / That's probably a good idea
rather than
Probably that's a good idea / That's maybe a good idea.

...where all four versions do occur, are perfectly valid, and mean the same. It's just that once a given construction has started to become "favoured" (perhaps by mere chance), it tends to become increasingly "idiomatically established".