Does one say, give your best or give of your best when it comes to effort?
Give your best
perform a task as well as possible; "The cast gives full measure every night"
"Give of your best", implies not only effort, but everything, so I would just use "give your best", when specifically referring to the one thing, effort.
One would usually say :
give your best.
"Give of your best" has the implication of giving all about yourself that's the best. For example, there is a hymn that goes "Give of your best to the Master", and goes on to tell about your strength, and your youth, and your intelligence, etc.
You could use either.
"Give [it] your best [effort]" is the more common of the two, and is the stronger statement as well. It is asking you to put as much effort in as possible.
"Give of your best [work/effort]" has a somewhat archaic feel, and tends to appear mostly in religious contexts these days. When talking about effort it has the same meaning as "Give your best", but when talking about something more tangible or countable it is slightly weaker; it is only asking you to select from among the best things, though it is strongly implied that the very best should be chosen.
Comment time insufficient, so read here:
In the latter usage, "of" and "from" would be synonymous: to give of one's best is to give from (that which is) one's best. In the first case, the thing given is part of the giver; in the second, a possession of the giver. Give one's best tends more to connote quantitativeness, whether literally or figuratively. Giving of one's best tends to take more of a qualitative connotation. It is certainly true that "of one's best" would be tend to be seen often in versical literature such as Bible scriptures.