We downloaded the space gardens mappack, mine works fine, but for him Steam crashes before we even get into the lobby with the message:

"Map_LoadModelGuts: Map with no texinfo, maps/mp_coop_lobby_3.bsp Engine Error"

Now we've checked and the only mp_coop_lobby map he has is mp_coop_lobby_2.
We've tried Verifying the game cache, and it decided he's missing one file, which it didn't sort out.

Any ideas?

I should point out that when it comes to downloading maps, I know what to do, but he hasn't got a clue.

There are definitely better experts here to help you, all I could guess would be this:

Delete all the folders he might have messed up by installing Space Gardens incorrectly and try to verify the game cache again then. It'll take way longer to download the missing files but it should take less time than a complete reinstallation.

lobby_3 is the updated lobby from the first DLC. Sounds to me like his game isn't completely updated, he should make sure it's got automatic updates on. If that still doesn't fix it, do as the person above said and delete the folder entirely and then verify the game files.

worst case scenario sent him your map
But verifying the game cache SHOULD do it, if it fails once and it's still not finished, try again

OK, fixed.
For some reason lobby 2 managed to get in the DLC folder.

Thanks anyway!