Learn Spanish (Latin America): La granja tuvo mucha lluvia ayer.

The word ayer (yesterday) gives us a clear, completed timeframe. Because we are talking about a finished event in the past rather than describing an ongoing background condition, we use the preterite tuvo rather than the imperfect tenía.

Is it normal in Spanish to say a place "had rain" (tuvo lluvia)?

Yes, it is! While you will often hear people use the verb llover (to rain)—as in, Llovió mucho en la granja ayer—using tener is also a very natural way to describe the weather that a specific area experienced.

I sometimes mix up muy, mucho, and mucha. Why is it mucha here?

Lluvia (rain) is a noun. When you want to say "a lot of" before a noun, you must use mucho, and it has to match the noun's gender and number. Since lluvia is feminine, it becomes mucha lluvia. You only use muy (very) before adjectives or adverbs (for example, muy lluvioso).