There is no certitude of the particular usage that is involved; the traditionnal meanings that this word takes on are those of "2." and "3." in the following (wiktionnaire).
sûrement \syʁ.mɑ̃\ invariable (orthographe traditionnelle)
- (Dans un sens affaibli et par antithèse exprimant l’incertitude) Probablement, sans doute.
Il a encore sûrement fait une erreur.
Je viendrai sûrement à ta soirée.
- De manière sûre ; certainement.
Il a sûrement bu la tasse en se baignant.
- Avec sûreté, en sûreté, en toute assurance.
De l’argent placé sûrement.
Vous pouvez marcher sûrement par là.
The first sense is confirmed to be extant by the reverso dictionary as sense "2".
The first sense (in the wiktionary, or the second in the reverso) is a controversial one; it's been extant for quite some time before the TLFi was completed and yet does not appear in it. It is not found either in the Larousse en ligne. Nor is there any trace of it in the dictionnaire de l'Ac.. This dictionary (Mediaco) does not know it. It's a problematic word or usage, if you prefer, because, as said in the explanation of the above definition the meaning is antithetical to that in "2.": as you use it that way you say the contrary of what you say by means of "2.". It is problematic because it is confusing; its use in sense "1." nullifies its sense "2." meaning and vice- versa. You may very well find yourself at some point in life, in particular if you do not read too much, that you've acquired a habit of using it in only one of the two senses, not really knowing the other, and being even confused as to what is really being said sometimes. This sense "1." is in my opinion a horror of a lexical definition and I support those dictionaries that do not mention this first sense or warn the users of its aberrant incongruity.
It is unlikely that a very young child will have acquired both and more unlikely that he/she will be using both. In this particular instance ("C'est sûrement maman. C'est sûrement le matin..." ) the thought leading to the child's utterance is brought about by a preliminary reflection about a possible doubt (Could it be something else?); in the semi consciousness of the instant that precedes waking up I think there is rather an inclination to make tentative assertions, not so much categoric ones such as "I have no doubt, that's Mom, that's morning!", unless the child had great expectations in reason of a special happening that morning; this seems so specially as in the human mind the incertitude between dream and reality has its utmost intensity at the precise moment of waking up time.
Given the undoubtable existence of the controversial sense "1" of the wiktionary in the population and given the circumtances (young child, very possibly a context of intense doubting) I tend to think that the child's usage is this controversial one. It is in any case not too convincing that a child would partake of the more literary use of « surement », which in daily life is render commonly in French by « bien » (C'est bien Maman, c'est bien le matin !).
However, I see no possibility at all to exact from these words a reproach to the mother, such as for instance "It's too early, Mom!". There might exist in the child's mind, along this probable impression of doubt, a feeling of aversion at the idea of getting up but that belongs to the rest of the text and has no incidence in the discussion.