The eyelet on the tip of one of my rods popped out and broke yesterday. Can a new eyelet be installed or will the tip have to be replaced?

You'll have to replace the tip top guide. It's a very quick and simple process though and won't effect the rod at all if done properly. 

New tip

Rod tips are usually put on with the same kind of glue used on arrow heads.  I just has to be heated and removed.  Then a new one glued in.  I had that happen to a rod this week and got it replaced for less than $5.

Thx

Gentle heat....slowly. it doesn't take much to melt a blank!

I've fixed many that wouldn't come off with heat. Had to carefully cut the tube with a Dremel and cutoff wheel and pry the tube off with a small screwdriver.

I'm having a local builder replace it.

I want a top that can handle braid with a leader knot. I also want the eyelet a size larger. Any Fuji model suggestions?

I want the best I can get.

BFAT-6 (if guides are black)

If silver, use CFAT-6