Adjectival modifiers follow the gender, number,and case of the noun they modify (except in some tricky cases when an adjective is used with a number and a noun). Этого is the Genitive form of этот
It isn't any more complicated than that. «У» will always take the Genitive case.
The Nominative forms of этот are этот (m), эта (f), это (n), эти (pl).
The Genitive forms are этого, этой, этого, этих.
All adjectival Genitive -ого /-его endings always have the [v] sound, as well as the endings of some pronouns that follow. Here are some examples of these forms:
- добрый → доброго
- мой → моего
- какой → какого
- он → его
- большой → большого
- the word сегодня ("today"), a combination of old-fashioned сей ("this") and день ("day")
This spelling is historical;it can be traced back to the times when Г represented a voiced fricative, similar to h in "ham", only voiced.
In бог ("god"), г is traditionally pronounced as х (though, all other forms of the word are pronounced as usual).
Without context, you can't tell whether 'the' or 'a' would fit best (or even no article - my Russian wife often adds articles in her English even where we don't require them e.g. "the Market Street"). Duolingo should usually accept either (and should have accepted your correct 'this', as 'этого' is present).
The Genitive form is required because the preposition у attaches nouns in Genitive. The most typical Russian structure of possession is built like a statement of existence. In essence, it is not much different than "In this room, there is a cat", only with a child instead of a room. У is the preposition used for the possessor in such sentences.
The form ребёнка is built this exact way because of the fleeting vowel in the suffix ёнок (онок). This happens in all nouns with this type (котёнок "kitten", волчонок "wolf pup" etc.).
You can see another example of a fleeting vowel with the -к- suffix as in кошка, девочка, ручка, спинка, спичка, лампочка, электричка, палка. These nouns lack a vowel in the consonant cluster in their base form. But they get it the Genitive plural, when there is zero ending (they become кошек, девочек, ручек, спинок, спичек, лампочек, электричек, палок)
(historically, this branch of Slavic languages had short vowels. They eventually merged with usual vowels or disappeared—in an alternating pattern. This is why some roots and suffixes magically lose or get о or е depending on the form)