Funny, I could have written this post!
Just celebrated our 10 year anniversary and I had a $3k budget for a right hand ring. We have three daughters, so I knew I wanted something pink. Got a three stone ring with a pink topaz in the middle and two small diamonds as accents. I love it and have been wearing it every day since I got it in July. I couldn't find anything I liked that was already made so the entire thing was custom. I used a gem cutter for the pink topaz and used Quest Jewelers in Farifax, VA to make the actual ring. Entire thing came in around $3100.
A big blue topaz would definitely be within budget. Topaz is very reasonable. Just be careful if you are hard on your jewelry, as it's one of the softer stones and can chip easily. My advice is decide on the color stone first and then do research (for example, in the pinks, there were so many that I never even knew existed, like Morganite, Spinel, etc.) Same for blue stones, green stones.
The other thing I considered was a Cartier trinity ring. It's three bands, one each of yellow, rose and white gold, symbolizing the past, present, future. Available in different widths and also with or without diamond accents. Well within your budget as well.
http://www.cartier.com/en/jeweler-watchmaker/cartier-trinity-ring
Good luck. What a fun project.