A word for someone who notices things very keenly

I think the set phrase to be onto something expresses the acute sense of noticing something, as only the word awareness can suggest:

in or into a state of awareness about

(Merriam-Waster)

Another good word for it, perhaps even better, is the phrasal verb to sniff something out:

to discover someone or something, usually only after a special effort

Perspicacious

According to Wikitionary

(figuratively) Of acute discernment; having keen insight; mentally perceptive.

Your mother has a perspicacious nature. You can modify your sentence to use this word.

In simpler words:

notice changes quickly

The word that comes to mind for me is spot:

... unfortunately she spots your attempt and says ...

In this context, spot means to notice something:

spot

VERB

[WITH OBJECT]

1. See, notice, or recognize (someone or something) that is difficult to detect or that one is searching for.
   ‘Andrew spotted the advert in the paper’
   ‘the men were spotted by police’

(Oxford Dictionaries)

Although, if she's anything like my daughter, her attempts to change the subject will be obvious... 8^)

I would use either very observant, or very aware, or possibly hyperaware. Each of these choices could fit in your sentence, with some minor modifications to the sentence appropriate to the usage. Here's a sentence, that also happens to be a true fact:

The children of alcoholics often become hyperaware of body language, noticing subtle cues that most people are unaware of. This awareness can act as an "early warning system" of shifts in mood, or potential behavioral problems.

You will notice observant also fits in that context. In your sentence, aware is better suited, but observant can be used.

Unfortunately, she was aware of your attempt and said: "Don't try to change the subject, you (sic) got to answer my question first."

Or

Unfortunately, she observed your attempt and said: "Don't try to change the subject, you (sic) got to answer my question first.