
As you might expect, the #3 slot is the one with the highest number of lifetime HRs and the highest single-season mark (we'll cover the latter shortly).
The shifting around of hitters in the batting order makes it clear that no individual is going to come close to his career totals when we look at any particular BOP. But it's interesting to note that it's Ken Griffey, Jr. who gives Babe Ruth a run for his money for most HRs while batting third. These records are artifacts of lineup construction as much as anything else.
Chaining back to the #2 slot for a minute, we see that Eddie Mathews and Alex Rodriguez turn up in the top 25 for that slot as well as for #3.
Turning to the the single-season leaders, we have a strange anomaly that is probably exclusive to the #3 slot. The single-season leader for most homers while batting third is Mark McGwire (69, in 1998). He's also third highest, with 65 (in the following year, 1999).
Note, however that he is not to be found amongst the Top 25 hitters on the lifetime chart. That's because before he was traded to the Cardinals, he rarely batted in the #3 slot; while he played with the A's, he mostly batted clean-up.

For the cleanup slot, the race is much closer, with nine hitters within 40 HRs of each other. One name here might be surprising: Albert Belle. The glowery Cleveland slugger hit 96% of his HRs (366 out of total of 381) while batting cleanup, which is probably the record. (We'll defer to someone like the indefatigable Tom Ruane to go the full distance on this.)
It's not surprising, however, that some eighty years later, that Ruth and Lou Gehrig, baseball's original 1-2 punch, are still atop the lifetime totals for BOPs #3 and #4. (Though, as you can see, another slugger who originated in Cleveland--Manny Ramirez--gave him a run for his money.)
As you'd expect, McGwire turns up on this list, ranked #10 with his total of 302.
There are seven hitters who hit 50+ HRs while batting cleanup. Hack Wilson hit all 56 of his 1930 total from the #4; Mickey Mantle hit all 54 of his 1961 total there, as did Alex Rodriguez in 2007. Jim Thome (yet another Cleveland slugger--has any team produced such a group of sluggers at one time as the 90s Indians) hit all 52 of his 2002 total from the cleanup slot. Johnny Mize hit all 51 of his from the #4 BOP in 1947 (detecting a pattern here?). Mark McGwire hit 51 (all but one of his 1996 total) there for the A's--our lone exception.

Gehrig doesn't make the 50+ list (he never hit 50 homers in a season), but he shows up four times in the list of 40+.
All in all, there are 51 player-seasons where hitters hit at least 40 HRs batting clean-up. By way of comparison, there are 74 player-season where hitter hit 40+ HRs while batting third.
We'll continue down the food chain on this one shortly...stay tuned.