I just got a new crankbait rod and the lure weight on it is 1/4 - 7/8. I realized today that most of my deep diving cranks are 1 ounce. How bad is it for the rod to go 1/8 ounce over the suggested lure weight?

Should be no problem at all. I regularly go 1/4 or more over with no ill effects. 

10 minutes ago, EvanT123 said:

Should be no problem at all. I regularly go 1/4 or more over with no ill effects. 

Thankyou. I used the 1 ounce cranks today. Nothing seemed to be different from when I started.

You will be alright. Just don't try to throw say a 10xd and you'll be fine

Lure ratings are just guidelines. Some rods are rated heavier than they are capable of and vice versa. You'll be able to tell if it's overloaded. 

It is funny one of my favorite rods is a Powell Max 683cef. This is an original not the Max 3D series. It is 6'8" long and is rated as a medium heavy casting rod with an extra fast tip. The lure rating is 1/4 tp 3/4.Tackle Tour did one of is reviews on this rod.  They tested it on their machines and then with three different reel/line combination. They said it did not cast a lure less than 1/4 well which was no surprise because the company said it was a 1/4 to 3/4 rated rod.  Now they started throwing every kind of bait above one quarter all the way up to an 8 inch swimbait which is clearly way out of its specs. They said it ate it up and performed flawlessly even with 2.5 ounces of swimbait tied on.  Not very many rods would do that without breaking, but it does show you that most rods can go a bit above their rating safely.

1/8 of an oz isn't very much over. You should be good to go. 

That'll be fine.