Why did Elwood trade a Cadillac for a microphone?

It's a joke to emphasise just how poor the Caddy, which we don't see on screen, actually was. He didn't even sell the car. He just traded it.

However, an explanation as to why he did it can be found in the next line in the script:

Jake: A microphone? (beat) Okay I can see that.
Elwood: This was a bargain. I picked it up at the Mount Prospect City Police auction last spring. It's an old Mount Prospect Police car. They were practically giving them away.

So he got a pretty good car in the end. And hopefully a decent microphone!

The joke is that music is more important to the Blues Brothers than anything else, even the Caddy. Something as paltry as a microphone is worth more than a car, because they value music as more important. Consider everything else they do to get the band back together, all the laws they break, and you will see that their priorities are screwy.

It's a joke, but it's actually quite reasonable. Some studio microphones sell for astronomical prices.

Here's the Cadillacs you can buy on ebay for less than $10,000. Swap one of those old clunkers for one of these (Neumann microphone worth $AUD23,849.90) and you've scored yourself a real bargain.

While emphasizing that the car was a hunk of junk, and considering that studio microphones are expensive, getting "a microphone" would seem to fit their slang for getting a gig or a show. They also appreciate music more than anything else. This does conjure up the idea of an interesting story in Elwood's past where he might have not had any control in the "trade". I don't think there's any explicit meaning in what he's telling Jake.

I think this is a bit of well-written setup dialog for the scene that comes immediately after the dialog in question:

Jake: Well, thank you, pal. The day I get out of prison, my own brother picks me up in a police car.
[Fog horn and warning bell sounds.]
Elwood: You don't like it?
Jake: No, I don't like it.

We all know what happens next. It's early in the movie, and you need to get Jake established a bit as a character. You're also seeing that Elwood is kind of an idiot - Jake is the smart one of the two. This dialog serves very well to lay that out, while also being hilarious.

I'll go with cde's answer (https://movies.stackexchange.com/a/47229/95051) that it's about priorities.

Trading the caddy for another car? No.

Trading the caddy for a microphone? Well that's different.

Also, I always thought that there was an implied subtext that Jake needed a microphone for a gig, or something like that, maybe because of a last minute equipment failure

And when push came to shove, getting the mic and doing the gig was more important than the original Bluesmobile.