Learn Russian: Мы не поехали туда из-за дядиной ошибки. - We didn't go there because of uncle's mistake.

Possessive adjectives with the suffix -ин agree in gender, number, and case with the noun being possessed, not the owner! The noun here is ошибка (mistake), which is feminine. Therefore, the adjective takes the feminine form дядина, which changes to дядиной to match the case of the noun.

Why do дядиной and ошибки take the genitive case here?

They follow the preposition из-за (because of), which always requires the genitive case. Notice that in the mixed declension of possessive adjectives, the feminine genitive ending is -ой, just like a standard adjective.

Could I use потому что instead of из-за to say 'because of'?

No, because потому что means 'because' and must be followed by a full clause with a subject and a verb (e.g., 'because the uncle made a mistake'). When 'because of' is followed simply by a noun phrase, you must use a preposition like из-за. Из-за is also especially common when talking about negative reasons or assigning blame.

Why use the perfective verb поехали when the action of going never even happened?

Negating a perfective verb of motion simply states the fact that a specific, one-time trip didn't happen (we didn't set off). If you used the imperfective не ехали, it would focus on the process, meaning 'we were not in the process of driving there' or 'we didn't use to go there'.