Learn Turkish: Kuzenimin bebekle ilgilenmesi lazımdı. - My cousin had to take care of the baby.

Because this sentence uses a verbal noun (ilgilenmesi). In Turkish, to say "my cousin had to do something", you literally say "my cousin's doing it was necessary." Since the cousin "owns" the action, kuzenim takes the genitive (possessive) ending -in.

Why is it "bebekle" (with the baby) instead of "bebeğe" or "bebeği"?

The verb ilgilenmek (to take care of / to be interested in) always pairs with the word "with" (ile or the suffix -le / -la). In Turkish grammar, you don't take care "of" a baby, you take care "with" a baby.

How do the suffixes on "ilgilenmesi lazımdı" build the meaning "had to take care"?

First, we turn the verb into a noun: ilgilen + me = "taking care". Then we add the 3rd person possessive -si to show who has to do it: "his/her taking care". Finally, lazım (necessary) gets the past tense suffix -dı (was). Put together, it literally means "His/her taking care was necessary."

Could we use "gerekiyordu" instead of "lazımdı" here?

Yes, you can say Kuzenimin bebekle ilgilenmesi gerekiyordu. Both mean the exact same thing. Just remember that lazım is an adjective, so it takes the simple past suffix (-dı), while gerek is a verb root that uses the past continuous tense (-iyordu).