At the risk of sounding a bit snobby I would like to state that Not all thread is equal.
I am seeing new thread dyers popping up all over the place and that is a good thing. However, I think it is important to educate those who are new to HDT so that you know you are getting the most for your money.
Be wary if the seller doesn't let you know in the item description what the source thread is. Many sellers are dying inferior threads with inferior dyes and selling their threads at the same prices as those of us who start with the very best materials money can buy.
You want to make sure your hand dyed threads are 6 cord rather than pearl cotton or other 3-ply threads.
6 cord threads are smoother, tighter, stronger, tat better and give you a superior end product. These threads are harder to come by and have to be special ordered. Whereas pearl cotton and other such threads are fuzzy, softer, and weaker and can be purchased cheaply at any hobby craft store.
As for the dye used you want to make sure the colors are colorfast and that the thread has been properly cured and washed so that it doesn't bleed out when blocked.
As and aside, here is an observation I have made: Good thread + a good pattern = no blocking. (Just something to think about) I very rarely have to block my work.
My feeling is that if you are going to spend hours and hours tatting beautiful lace for friends and family why not start with the very best materials in the first place?
If you are going to spend the extra money on HDT then make sure you know exactly what you are getting. The price of the thread should reflect the materials used as well as the expertise in creating the product.
So with that said, if you so choose, feel free to buy those hand dyed pearl cottons as long as the price reflects the materials. Just make sure you aren't paying the same price for it as you would for the superior threads.