It's have become.
That's because it occurred at an uncertain point, it's not finite, and it still has relevance to the present. And, to be honest, while it is technically OK to shift tenses in writing, the rest of the sentence in the simple present (have got is simple present tense).
Although present perfect usually describes an action that has already occurred, it's still one of the present tenses.
I'd just like to know if replacing have become with became would still
be grammatically acceptable? What changes could be done to make became
Actually, when you replace have become with became, you drastically change it's meaning. When you use the simple past tense, you are signifying to your reader/listener that this event is over, and in this case, that the trend is no longer progressing.
I think [simple present] every one of us needs [simple present] a
cellphone. Cellphones have become [present perfect] part of daily
life and I've always [signals ongoing] got [simple present] mine
I don't have a problem with tense shifts, but I've marked up the above sentences to demonstrate the other reason that present perfect is a better choice. In those two sentences, you have three other present tense verbs. The simple past would really stand out.
Now if we were talking about an event that's over, it'd be OK to use became. I can tell just from the context that that's not what you intended to say. You mean to say that the trend still has relevance (and you're not exactly sure when it started).
That's the difference.