Short answer: What Barnidge did last season was a fluke. If you look up the meaning of fluke in the dictionary it says see Gary Barnidge 2015 season.
As Anarcy and others have already described, there were a confluence of circumstances that all aligned themselves to his opportunity and performance last season.
Cleveland did not have many other options to throw to besides Barnidge, Benjamin and Johnson. The only WR threat was mainly a field stretcher, so this leaves the easy dump off options to the TE and RB.
He averaged 7.8 targets per game. That is pretty much the ceiling for the top TE in the league. Gronkowski has only had more than 125 targets once in his career thus far. Antonio Gates once. So his opportunity was near the top for elite TE players, yet he is Gary Barnidge. This was 20.5% if the total targets.
The Browns had 1042 offensive plays last season and they threw the ball 609 times. Hue Jackson offenses average 1000 offensive plays and the run to pass ratio is very balanced, which means around 500 passing attempts. The most he has had a QB throw was 524 times with the 2011 Raiders and a combination of Carson Palmer+Jason Campbell. So even if nothing else changed and he maintains a 20% share of the targets (which is doubtful) that would be 100 targets.
Most of his good games came against
weak opponents. and when McCown was starting. Games 3, 4 and 5 were against Oakland SD and Baltimore whe were very bad against the pass last year. He had 10 targets in two of those games, a lot for any player. Game 6 against Denver he had 9 targets but only caught 3 of them. This gives an indication of how well he does against a capable secondary, not nearly as well. This is pretty much the pattern for the whole season. The Browns just had few other options besides him last season. That is not the case this year.
Tyler Eifert only has one decent season of data to draw on which was 2015 where he played 13 games. He averaged 5.7 targets per game which would be 91 targets over 16 games. This was 18% of the targets. Barnidge is not Eifert though.
I see Barnidge getting 16% of the Browns 500 passing targets in 2016 which is 80 at a 62.5% catch rate (last 3 seasons for Barnidge) that is 50 receptions. At 13.4 ypc that is 670 yards which seems generous and perhaps 5-6 TD scored.
Those numbers were good for TE 9-10 last season in standard scoring leagues and about TE 12 in PPR leagues last year.