Well my opinion is, it's a home game, I'm assuming you're somewhat friendly, or at least feigning friendship at least(seen as you said you want to keep him because he is weak). A quick way to ruin a 'friendly' home game is to be strict on certain players, because you don't know how the other players will react to that either. My point is you could be strict and start calling the clock on this player, but maybe another player or two might not like that either, etc.
Now I ran a home game for years, and had a few players who were extremely slow, very similar situation to what you've described. My solution, friendly ribbing, as in a humoured, light hearted insult whenever it was on them. Nothing over the top, something that got a laugh but still got the message across to the player to not take 5 minutes.
My point is often home games have a bit of a looser feel with rules, and often you'll get players who only play home games because they're more relaxed and not as strict with rules. You know the mood, players and atmosphere of your game best so perhaps this isn't the answer but just a suggestion to try keep it friendly, well unless every other player at the game is super serious.