A pair of the most overlooked, marginal positions in the NFL occupies the special teams unit. Punters and kickers are often overlooked when comparing separate teams, but these positions are easy to upgrade, and they are often the difference in tight games.
For the reasons listed above, I think one of the places a team should begin to rebuild is with kickers. You can normally get franchise punters and kickers in the late rounds of a draft, and their contracts are very cheap for a considerably long time.
Last year, the Jaguars, who are now in full-blown rebuild mode, drafted punter Bryan Anger in the third round. The team didn't get top value out of the pick, but they picked a safe player that could be with the team for the next decade and a half. Anger had a great rookie season, and he solved one position for the team at a relatively low price.
I'm not advocating drafting punters and kickers in the top few rounds. I do, however, think a late round pick (5th, 6th or 7th rounds) is better spent on a kicker that will play immediately instead of a scout team player that may be cut halfway through their rookie season.
Punters and kickers are relatively easy players to evaluate coming out of college. The kicking game doesn't drastically change from college to professional football. There's no increase in player speed to adjust to, and it's a matter of adjusting to slightly different footballs. Picking a great kicker in the draft is about the safest pick you could make, and teams see returns on that investment for many years in most cases.