Dear 100 Hour Board,
Who said "the [best] convert of [one's] mission is [himself/herself]" (roughly paraphrased)?
I feel like Sunday Night Banter is spot on, and it's either something Elder Holland came up with or something that was created in the missionary department that he said to the missionaries and popularized the phrase. Either way, it seems to be a phrase that missionaries (or anyone who brings up missionaries, for that matter) love to use, especially in conjunction with the DC 18:15-16 quote about how "if it so be that you should labor all your days in crying repentance unto this people, and bring, save it be one soul unto me, how great shall be your joy".
I think that this is a great quote, depending on the circumstance. I personally feel that though I definitely deepened my conversion on my mission, there are definitely people whose change in spiritual direction was much more drastic than mine, and I would say that their conversion was "better than mine." In this regard, I like the way that Elder Holland phrases it in Sunday Night Banter's link: "Missionaries are under obligation to come home having had at least one convert, you!" Either way, if you look up <"greatest convert" "missionary"> on Google, you'll get a lot of great missionary blogs.
Thanks for the question,
Dear Thomas Jefferson,
Your question is a little tricky because it's nearly impossible to tell who the first person on God's green earth said this, but I know that Elder Holland believes it and has said it. You can read a Church News article to find out exactly what he said and who he said it to. We know that in 2004 this phrase was widely known, so it's a pretty good guess that Elder Holland wasn't the first person to articulate this thought.
I hope this helps.
-Sunday Night Banter
There is also this quote from President Nelson earlier this year: "His or her most important convert will be himself or herself." He also said it in 2017. I've included it here because even though the earliest reference we could find was from 2013, I find it hard to believe that the phrase/sentiment originated even in the past ten years. If I'm right, it is interesting that, as far as we can tell, it has become more widely used/taught by general authorities in recent years.